How can I make cookies with no eggs?

There's a replacement, but it's not an ingredient most people have sitting around -- flax seed.

I use it when I'm cooking for a few of my friends who are vegans. Grind up some flax seed in a coffee grinder. For each egg, take 1 TB of flax seed meal, and 3 TB of water. Stir it up and let it sit for a while -- it'll get kinda slimy. Use that in place of the eggs.

Obviously, this won't work for anything where the whites and yolks are used separately, but I've used it for years for Christmas cookies -- the only problem is that the dough doesn't hold together quite as well when you're dealing with refrigerator cookies.

update: it's also a good idea to make friends with your neighbors. I've known the folks living in one of the houses behind me since high school ... we've been known to call each other up when we're short on sugar / flower / eggs / milk / vanilla, etc.

I've cooked a number of batches of cookies based on small variations of this recipe, which is very simple and requires no eggs. I generally use a little more oil and water than the recipe recommends, but perhaps that's just my preference.

The first thing I thought of when I saw this question the Tool song "Die Eier Von Satan". Its lyrics are a recipe for cookies that have no eggs in German.

After reading the recipe in English, it had occurred to me that it is probably not far off a recipe for shortbread. I realize that you had asked for substitutes for eggs, but maybe a better alternative is to make a different kind of cookie. Shortbread cookies are awesome and tasty and the recipe is simple. ...and no eggs!

From allrecipes.com

* 2 cups butter
* 1 cup packed brown sugar
* 4 1/2 cups all-purpose flour

Directions

1. Preheat oven to 325 degrees F (165 degrees C).
2. Cream butter and brown sugar. Add 3 to 3 3/4 cups flour. Mix well.
3. Sprinkle board with the remaining flour. Knead for 5 minutes, adding enough flour to make a soft dough. Roll to 1/2 inch thickness. Cut into 3x1 inch strips. Prick with fork and place on ungreased baking sheets.
4. Bake at 325 degrees F (165 degrees C) for 20 to 25 minutes.

I just make no-bake cookies when I want a quick dessert with no eggs. There are a number of recipe variations available with or without peanut butter, marshmallows, oatmeal or coconut, cocoa or chocolate chips. I usually have the ingredients in the pantry for some version, and the nice thing is that I don't have to turn the oven on and heat up the kitchen - however these cookies do not travel well here in the American southwest as it is hot enough outside (currently 111 degrees F) to remelt them and you just end up with a bag of melted cookie goo.

This is a pretty good page of vegan egg substitutions.

You can substitute canned pumpkin for eggs.

I have a very simple recipe for egg less almond cookies...

**Ingredients:**  
1.5 cups whole wheat flour / atta  
3/4 cup sugar (powder coarsely if using regular white sugar)  
1/4 teaspoon salt  
1/2 cup sliced almonds  
3/4 tsp green cardamom seed coarsely powdered  
3/4 cup unsalted butter, softened to room temperature  
About 2-3 tablespoons of milk (only if you need it) 

**Method:**  
1. In a bowl, mix the flour, sugar, salt, sliced almonds and cardamom powder well.  
2. Next add soft butter (and milk if needed) to the flour mixture to make dough.  
Dough should be very soft.  
3. Divide the dough into small balls. Press each ball between your palms  
lightly; every piece should be about 0.5 inch in thickness. Place the dough balls  
on an ungreased cookie sheet or parchment paper about an inch apart.  
4. Pre-heat oven to 360F / 180C. Bake the cookies for about 18-20 minutes or until  
cookies are lightly gold brown. After they become lightly golden brown remove the cookie sheet from the oven.  
Let the cookies cool down for 5 minutes before taking them off the cookie sheet.

Note: Make sure that the butter is completely soft and at room temperature, otherwise you will get crumbly dough when you mix and end up adding more milk than necessary, which will make the baked cookies more crackly and hard.
These are absolutely delicious and the flavour of cardamom makes it extra special.