There is only one egg used to make the egg wash and it is brushed on right before baking.

I haven’t tried a dairy-free or vegan version myself, so I can’t promise how it’ll turn out—but here’s what I’d do. Use a vegan puff pastry (many are naturally dairy-free), swap the butter in the filling for a plant-based option like Earth Balance or Miyoko’s, and replace the egg with 3 tablespoons of unsweetened applesauce or a flax egg (1 tablespoon flaxseed meal + 3 tablespoons water). For the egg wash, either skip it or brush with a little plant-based milk to help with browning.