If you want to keep your baking purely vegan you can replace eggs with “chia eggs” or “flax eggs.”
For the equivalent of 1 egg use 1 tbsp of ground chia or flax and 3 tbsp of water. Mix those together and let sit for a few minutes. It will turn gelatinous. (So 2 eggs would be 2 tbsp of ground chia or flax and 6 tbsp of water).
This will work exactly like eggs in your dish without changing the taste at all. Plus you get the added benefits of the chia or flax!