What can I use to replace eggs in the recipe?

It looks like you're basically making a cheese custard for the center of the bread. Flax seeds aren't going to give you a firm, egg-like texture. Instead, I'd use something that already has that cooked-egg texture, like tofu, which also won't impart any flavor (so you'll need to tackle that issue separately).

Experiment with different preparations, but I'd start off with a silken tofu, then purée the tofu in a blender with a bit of cornstarch (to help it firm back up when cooked). Once the tofu has been processed you can mix in the shredded cheese. and throw the filling in. You'll probably have to do several attempts before you get the ideal preparations and keep in mind that it won't taste like the original thing, but may bear enough resemblance.

EDIT: In response to the comment, I would say you could try using tapioca in a high concentration, dissolved in a bit of milk or cream. After cooking (hopefully) the tapioca will help to create the texture you want, and it will actually not affect the flavor.

The other (obvious) option is to just leave out the eggs and not replace them with anything. It won't be the same, but neither will any of these substitutions, and it will likely still be good.

Eggs have various properties, among them:

• as a "glue" as you suggest, you can replace them with some sort of gelatin or pectin. You can find it naturally in some fruit seeds or fish/chicken bones. Industrial flans use agar-agar from red algae.

• as an emulsifier, as in mayonnaise, you can replace them with garlic, mustard or anchovy. Many mediterranean recipes use this (e.g., aioli).

This question provides several generic vegan egg replacements. One that may suit your needs is flax seed and water.

Also, while you mention that you can't take many chicken eggs, how about replacing with another egg? Does that work for you?

I'll post a few of the example I feel would work well for your situation below, but the following page has some great substitutes for eggs in different situations you would use them.

http://www.foodsubs.com/Eggs.html

For Filling you may want to consider using Silken Tofu as it somewhat resembles the texture of a scrambled egg and takes on flavors very well.

If you'd like to substitute the eggs in the dough you could use a couple things:

• 1 egg = 2 tablespoons liquid + 2 tablespoons flour + ½ tablespoon shortening + ½ teaspoon baking powder + 2 drops yellow food coloring if you want the color from the yolk
• flaxmeal (Make flaxmeal by grinding flaxseed in a blender until it has the consistency of cornmeal. Use two tablespoons flaxmeal plus 1/8 teaspoon baking powder plus 3 tablespoons water for each egg called for in recipe.
• mayonnaise (Substitute 3 tablespoons mayonnaise for each egg called for in recipe.)
• gelatin (To replace each egg: Dissolve 1 tablespoon unflavored gelatin in 1 tablespoon cold water, then add 2 tablespoons boiling water. Beat vigorously until frothy.)

There is similar question and it has accepted answer With what can I replace eggs? . I can't comment so I added it this way.