Cheesy Hand Pies

Whisk 2 1/2 cups flour and 2 tsp. salt in a large bowl. Grate butter on the large holes of a box grater right into bowl, tossing occasionally with your hands to prevent sticking. Drizzle 3/4 cup ice water around circumference of bowl. Mix with a fork, adding more water if needed by the teaspoonful, until a shaggy dough forms.
Turn out dough along with any dry bits onto a clean work surface. Knead until dough is smooth and hydrated with only a few pieces of butter still visible, about 1 minute. Form dough into a disk, wrap tightly with plastic wrap, and chill at least 1 hour.
Place racks in upper and lower thirds of oven; preheat to 350°F. Whisk eggs in a medium bowl until well blended. Transfer half of eggs to a small bowl; reserve as egg wash. Add feta to remaining eggs in medium bowl and mash with a fork until combined and nearly smooth. Gently fold in mozzarella and dill until just combined; season with pepper and remaining 1/4 tsp. salt.
Divide dough in half. Roll each piece on a lightly floured surface to a 16x8" rectangle. Using a bench scraper, cut each rectangle into 8 squares (for a total of 16). Transfer to two parchment-lined rimmed baking sheets and chill 10 minutes.
Brush edges of squares with reserved egg wash. Mound a heaping tablespoon of cheese filling in the center of each square. Fold dough over and press edges with a fork to seal. Brush with egg wash, then cut 3 small slits into the top of each. Bake pies, rotating pans from front to back and top to bottom halfway through, until golden brown and crisp, 30–35 minutes. Let pies cool 10 minutes before serving.
Do Ahead: Dough can be made 3 days ahead. Keep chilled. Let rest at room temperature 15 minutes before rolling out.