Melt butter in a medium frying pan over medium-low heat. Add the apples, cinnamon, and sugar and cook, stirring occasionally, until they just start to become transparent, but not enough that they become mushy (about 5 minutes).

Remove from heat and let sit for 10 minutes.

Stir lemon, zest, walnuts, and cheese into the apple mixture. Pour apple mixture onto the center of the crust and spread it out so that there is about a 1” border around the edge. Fold up the edge over the apples, crimping as needed to go around the circle, creating a crust.

Whisk egg with a splash of water and using a small brush or your fingers, spread some around the edge of the crust. (This will give it a golden brown look.) You won’t use all the egg.