Step 1: Prepare your pie crust. Roll out one disc of pie dough and carefully fit it into a 9-inch pie dish. Trim the edges, leaving about a 1/2-inch overhang. Place in the refrigerator while you prepare the filling.
Step 2: In a large bowl, gently combine the cubed peaches with lemon or lime juice, 3/4 cup sugar, tapioca starch, vanilla extract, and smoked paprika. Stir until the peaches are evenly coated.
Step 3: Pour the peach filling into the chilled pie crust. Dot the top of the filling with the small pieces of unsalted butter.
Step 4: Roll out the second disc of pie dough. Cut it into 1-inch wide strips to create a lattice top. Carefully arrange and weave the strips over the filling, then trim the excess dough. Crimp or flute the edges of both crusts together to seal.
Step 5: In a small bowl, whisk together the egg yolk and milk to create an egg wash. Brush the lattice top and crimped edges with the egg wash, then sprinkle evenly with the remaining 1 tablespoon of granulated sugar.
Step 6: Preheat your oven to 400°F (200°C). Place the pie on a baking sheet (to catch any drips) and bake for 20 minutes. Reduce the oven temperature to 375°F (190°C) and continue baking for another 30-35 minutes, or until the crust is golden brown and the filling is bubbly. If the crust browns too quickly, cover the edges lightly with foil.
Step 7: Once baked, remove the pie from the oven and let it cool completely on a wire rack for at least 2-3 hours before slicing and serving. This allows the filling to properly set.