Over medium heat, add 1 tablespoon olive oil and sautee shallots for a few minutes until softened. Add chopped greens and saute for 2-3 minutes until wilted and add to a bowl and cool for a few minutes.
Add the feta, mozzarella, dill and lemon zest to the greens and mix to combine. Taste for seasoning and adjust as needed.
Roll out phyllo and place a damp towel over the sheets not being used yet.
Place 1 sheet of phyllo on a clean surface and brush with olive oil all over. Repeat with 2 more sheets, stacking them on top of each other.
Take about 1/2 cup of filling and place filling along side the longer end of the phyllo. Roll phyllo up so it looks like a long snake and then curl one side in, so its coiled into a round borek.
Continue with remaining phyllo and mixture until you have 6 borek.
Whisk egg wash and milk together and brush egg wash all over borek and sprinkle with zaatar, Aleppo pepper and an extra dusting of grated Parmesan cheese.