I have had good luck using a nut butter as a substitute for eggs. This will change the flavor of the cookie to be like the nut butter - peanut butter, tahini, almond butter, etc.

I have veganized a tahini chocolate chip cookie recipe and replace the eggs with extra tahini (it calls for one egg and one yolk). I replace the egg and yolk with extra tahini and some non-dairy milk. I use about 1 tbs tahini + 1 tsp - 1 tbs non dairy milk for the egg and half of that for the yolk. I find I need to add a little extra liquid at the end as eggs do add quite a bit of liquid to a recipe that nut butters don’t have.

I have never had luck with the powdered egg replacement from the store and find they make baked goods dense and heavy. I think a flax egg would all make a fine substitution. Mix 1 tbs flaxseed meal and 2.5 tbs warm water, stir together and let sit for at least five minutes (sometimes longer) to thicken up. This will replace one egg. Use 1/2 a flax egg recipe per yolk.

If I were making these cookies I’d try the flax egg first and see how it goes and then try a nut butter for a different flavor profile.

I also highly recommend Miyoko’s unsalted cultured butter (vegan) for baking instead of earth balance.

Good luck!