Vegan – Best Egg Replacements

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.

Egg yolks and egg whites have very different roles in baking. You can almost never replace them. And in the cases in which you can, you will end up with a different texture.

In this case, it won't be sure disaster to replace, but I would be very reluctant to do it. Flourless recipes are finicky. Flour holds stuff together. Nut flours don't hold anything together, they need a binding agent. Egg whites are a good binding agent. Egg yolks are not only not good as a binding agent (except in certain circumstances, for example in custards - but you don't have this here), they are even a lubricant and as such will interfere with the binding done by the egg whites.

If you substitute here, you will not only change the taste, you risk your cookies crumbling apart in your hands.

In general, don't replace stuff in baking recipes. They are hard to design right. Even if you have some issues (e.g. allergies), it is easier to find an existing recipe without the offending ingredient than to try to tweak an existing one to work with different ingredients. Eggs are especially hard to substitute. Such a reason like "not wanting to waste" loses its sense if what you end up with is bad cookies, which are a much worse waste of products. And besides, you can do many more interesting things with a yolk than just scrambled eggs. In fact, I am frequently throwing out whites because I have found no use for them after doing something with the yolks.

If you still want to go ahead and try if the cookies work for you, try using 3 whole eggs. 2 eggs are nowhere near enough to substitute for 4 egg whites. Not only are the yolks smaller than the whites volume-wise, they will also not deliver you the binding proteins you need so badly.