Are there any vegetarian-friendly egg substitutes that can be used in cakes?

Yes, there are multiple substitutes. I can't explain it myself any better than here, but I have tried to summarize it:

The best substitute for egg in a dish depends on the dish itself (because the function of the egg isn't the same in every dish). As a general rule, the fewer eggs a recipe calls for, the easier they will be to substitute. Also consider how the substitute will affect the overall taste of the finished dish.

Possible substitutes:

• Bananas and/or apple sauce (not flavourless, baked goods such as muffins, pancakes or yeast-free quick breads, especially banana bread; if you want something fluffy, make sure to add a bit of baking powder)
• (Silken) Tofu (flavourless, best* substitute for quiche, fritatta or egg salad, can work in 'heavy baked goods' such as brownies or pancakes, doesn't work well in fluffy dishes; add bit of mustard, turmeric or nutritional yeast if you want the same colour as you would have used real eggs; I advice you to search for recipes that use tofu in stead of egg in the first place, as other recipe adaptations can be required)
• Ener-G egg replacer (flavourless, works best* in baked goods, it can also act as a binder; if you chose to use a different brand, be careful to read the ingredient list, as some may contain egg white)

Less common substitutes (though I'm not sure if/how they work):

• Flax seeds
• Soda
• Bread crumbs or oats (for savory dishes)

*) In the opinion of the site, I have no experience with it.

There are substitutes, but eggs have very important effects on the texture of dough and are therefore hard to substitute. You will need to experiment a great deal until you hit on the correct texture, because you'll need to tweak all other ingredients and maybe include new ones, e.g. substitute part of the butter and use cream instead to account for moisture loss from egg substitutes.

Mien's link already includes some substitute ideas, like applesauce and commercial imitations. I can't give you more suggestions, because I always put eggs in my cakes ;). But when tweaking, keep following in mind:

Egg whites provide a resilient mesh which traps small air bubbles, so they make a cake more airy. You may need to adjust the baking powder upwards a bit. If it is a soda recipe, you'll need to raise the acidity too. But the whole effect can't be approximated just with baking powder, because its bubbles are too big. For that, you need better emulsification (covered below).

Egg whites also have a binding quality. This shouldn't need compensation in a cake, as you need a dough without too much binding (as opposed to bread), but if your cake starts falling apart, consider using a flour with higher gluten content (all purpose flour instead of cake flour). Gluten binds too.

If, on the other hand, your cake is too bread-like, then you may need less binding. The egg yolks are a source of fat, which prevents the dough gluten from forming a strong mesh by covering the particles. You will miss this with egg substitutes, especially if they are pectin-based (like applesauce) because pectin enhances binding. Use the lower gluten dough (it it doesn't cause the cake to fall apart) or add more fat.

Then there is the importance of emulsifiers. Egg yolks are one of the best sources of lecithin in the kitchen. An emulsifier helps your wet ingredients bind in an emulsion, instead of staying in two separate phases with the watery phase soaking the flour through and the fatty phase making the whole thing greasy. It also helps the trapping of very tiny air bubbles in the wet part of the dough (this is what happens when you beat eggs with sugar on high speed until the colour lightens) which provides a very fine leavening. Maybe commercial egg imitates contain emulsifiers - check the list - but if they don't, you must buy pure lecithin. It is usually soy derived, but maybe you need to check with the manufacturer that your brand is suitable for vegetarians. Also, consider including cream, the fat in it is already emulsified in water, unlike butter (which is water emulsified in fat). You must calculate precisely and tweak other ingredients to ensure that the total amount of fat and moisture don't change (except if you are compensating for the changes caused by the egg substitute).

If it is a chocolate cake, consider reducing the fat and the flour and adding some cocoa powder (the regular-fat variety). Cocoa has binding and thickening properties, and will help with the more even fat distribution. Melted chocolate tastes good in cakes, but makes them very dense, so don't use it.

The easiest way would be to find a recipe published by someone who has already done all the work. But just from paging through a cookbook, you can't tell if they did it right, or just slapped a spoon of applesauce instead of an egg and called it a vegan cake.

Edit: I just had an idea when I saw Mien's gelatine question and yours together in the list. I haven't heard of anybody using it, and don't know if it will work, but it does, it will be great. So try it.

I already said that using cream would help. As you need a foam full of very tiny bubbles for the perfect leavening, you could try whipped cream instead of whipped eggs. The trouble is that the air will get out of the batter very quickly while and after folding the cream. So you'll need to stabilize it. Gelatine in unusually small quantities will make for a very nice experiment, but obviously not one you'll want to do. But plant-derived gelling agents could help. Carrageenan is commonly used with dairy products, and you can buy it online. So try folding stabilized whipped cream into the batter.

There is a product called EnerG Egg Replacer. It is a powder binder/emulsifier that is specifically for use as an egg substitute in baked goods. You can find it at health food stores and sometimes in the vegetarian aisles at grocery stores.

For vegan scones you are looking to eliminate a couple of animal by products.

Firstly: milk

Any nondairy milk will suffice (that I've found) though the heavier ones, like coconut or hemp, tend to give you a better texture and consistency when done.

Secondly: butter

There are a lot of non dairy butter replacements that can be found at most grocery stores and chemically/structurally give you everything you need.\

Third (and as your question asks): eggs

There are a few ways to replace eggs when baking.

Here is a great list that includes what it's best to use it for:

1. Unsweetened applesauce; Muffins, brownies, cakes, bars
2. Plain or vanilla soy yogurt; Muffins, cakes, bars
3. Tofu, add to blender and add enough water to make smooth; Pies, quick breads, muffins, and cakes that are more dens
4. Pumpkin puree canned; Quick breads, muffins, cakes
5. Mashed banana; Quick breads, muffins, cakes
6. Soy flour; Muffins, cookies, cakes
7. Ground flaxseed; Brownies, cookies, and breads
8. Ener-G egg replacer; Multipurpose baking

Bold denotes methods I have used successfully. That said, the applesauce, pumpkin and banana change the flavor of your baked goods, making Ener-G the best replacement in most baked goods.

I love to bake, but can't eat eggs. I've had no trouble replacing the eggs in all of my favorite recipes. I use one of the following substitutions for each egg in the recipe:

ONE: - 2 tbsp. flour - 2 tbsp. warm water - 1/2 tbsp. butter - 1/2 tsp. baking powder

TWO: - 2 tbsp. ground flax seed - 1/8 tsp. baking powder - 3 tbsp. water

For each, combine in a separate bowl and let sit for a few minutes while you gather the rest of your ingredients. Then add as you would eggs.

Both of these subs can be found here: http://www.foodsubs.com/Eggs.html among others. These are the ones I found worked the best though.

The only baking I still use eggs for are angel food cakes and creme brulee... but you just can't mess with those!