Great question. My friend's daughter is newly allergic to eggs. She mentioned she uses flax seed in many recipe's needing egg such as brownies and sweet breads. It sure wouldn't work for making omlettes though, I think you might be out of luck for a direct egg substitute.

I've read about using flax seed as an egg replacer, but never tried it. I have successfully used plain or vanilla yogurt to replace the eggs in a muffin recipe (1 tablespoon of yogurt per egg).

I'm not sure if you can use anything in place of eggs in an omelet, but you might try making your daughter scrambled tofu when other folks are having eggs (assuming that she isn't also allergic to soy). As for whipped toppings, I've seen recipes using soft tofu or coconut milk, but haven't tried them (yet).

i would suggest checking out vegan sites, as they have omelette, and cake suggestions

The problem with finding a good egg substitute is the versatility of egg. Egg works as a binder, leavening agent, emulsifier, tenderizer, shortener, among other things. As a person with egg allergies myself, I've experimented with a few substitutes, and below is what's worked for me.

As a binder:

1 egg === 1/4 cup miso. I've used this successfully to bind burgers and meatballs. Reduce salt in recipe because miso tends to be salty. Isn't really suitable for dessert.

1 egg === 1/4 cup firm silken tofu. I've used this successfully to make a new york cheesecake recipe. No one was able to tell it was egg-free. I've also used silken tofu to substitute for egg in no-bake mousse recipes. I've used this for making quiche and omelets (doesn't really taste the same though, since egg is a major ingredient in both). Blend the tofu until it's creamy before using it (add a 1/4 tsp turmeric when making quiche or an omelet to give it a yellow hue).

1 egg === 1/4 mashed ripe banana. Works for banana bread and muffins. It will lend a banana flavor to whatever it is you're baking.

As an emulsifier:

1 egg === 1 tbsp flaxseed + 3 tbsp water. Microwave the flaxseed/water combo for 10 seconds, then blend until the flaxseed it thoroughly pulverized (I use a stick blender). You'll end up with something that resembles a beaten egg. I've used it for making eggless mayo and it tasted great. The only problem is that you have small bits of flaxseed in the mayo, but it's purely an aesthetic problem and not a flavor problem.

Purely as a leavening agent:

No success so far. I still have yet to find a way to make eggless angel food cake, meringue, soufflé, or even a fluffy sponge cake.

You mentioned brioche, 1/4 cup half and half (per egg) might work, although I haven't tried this. You could also use the half and half (or milk) as a substitute for the egg wash.

Also, I've tried those commercial egg replacements once, no luck there.

In general, vegan sites are a good source for finding how to substitute egg in any particular recipe (you don't have to follow the vegan recipe). One site I particularly like is http://vegweb.com/ , or just google vegan and the recipe you're looking for.