In a mixing bowl, add the all purpose flour and cold cubed butter. Use two forks or a pastry cutter to distribute the butter evenly into the flour. It should look like coarse crumbs. Add the sourdough discard and use a spatula or wooden spoon to mix until combined. Use your hands to gently pat the dough together. Wrap with plastic wrap and place in the fridge for 15 - 20 minutes.
While the dough chills, make the filling. Chop the kimchi very fine. Use your hands to squeeze out extra moisture, which can make the crust soggy. Mix together the kimchi, cheese, and peas. Add salt and pepper to taste.
Preheat oven to 400°F and line a baking tray with parchment paper.
Remove dough from the fridge and place on a lightly floured surface. Flour the top of the dough and your rolling pin. Roll the dough until about 1/8 inch thick. Use a 2 - 3 inch wide cutter or glass to cut out circles. Gently re-roll the scraps and repeat.
Arrange half of the circles on a baking sheet, about 2 inches apart. These will be the bottom of the hand pies. Place a heaping tablespoon of filling in the center of the circles. Then, take the remaining circles and place on top. Use your fingers and press all around the edges. Then, use a fork to press down and crimp the edges.
Make the egg wash by beating together the egg and milk. Brush the tops of the pies with the egg wash. Cut slits into the top of the pies to release steam.
Bake for 25 - 30 minutes until golden brown. Remove from the oven and let cool for a few minutes, and then serve warm.