Prepare two rounds of pie dough using a preferred homemade recipe, or use two ready-made pie dough rounds.
Refrigerate the prepared dough until thoroughly chilled. The dough may be prepared up to one day ahead and kept refrigerated until needed.
Peel the ripe yellow peaches, then cut the fruit into approximately 1-inch cubes. Cubing helps the tender, juicy peaches retain more structure than thin slicing.
Place the cubed peaches in a large mixing bowl. Add the freshly squeezed lemon or lime juice, granulated sugar, tapioca starch, vanilla extract, and smoked or regular paprika. Cinnamon may be used in place of the paprika if desired.
Fold the ingredients together gently until the peach pieces are uniformly coated and all components are thoroughly incorporated.
Allow the mixture to stand for a couple of minutes so the flavors can mingle.
Preheat the oven to 400°F (200°C).
Lightly flour the work surface, then roll one chilled round of pie dough into a circle approximately 12 inches (30 cm) in diameter.
Transfer the pastry to a 9-inch pie dish and carefully press it against the base and sides of the dish.
Spoon the prepared peach mixture into the crust, including all of the accumulated juices remaining in the mixing bowl.
Remove the remaining pie dough round from the refrigerator and roll it into another 12-inch circle.
Using a sharp knife, divide the pastry into 6 strips, with each strip measuring approximately 2 inches wide.
Arrange the strips across the filled pie, weaving them over and under one another to create a lattice pattern.
Trim away excess pastry around the perimeter. Press the lattice ends into the lower crust, then flute or crimp the edge to seal the two crusts together.
Stir the egg yolk together with 1 tablespoon of milk to prepare the egg wash.
Brush the exposed pastry evenly with the egg wash, then scatter 1 tablespoon of granulated sugar across the crust.
Cut the unsalted butter into small cubes and distribute the pieces over the exposed peach filling between the lattice strips.
Set the assembled pie on the center rack of the preheated oven. Bake for 40–50 minutes, until the crust is golden brown and the peach filling is visibly bubbling.
If the pastry begins to brown excessively before the filling is ready, loosely shield the pie with aluminum foil.
Once the filling begins bubbling, continue baking for an additional 3–5 minutes.
Remove the pie from the oven and leave it at room temperature for 2–3 hours before slicing. This cooling period allows the peach filling to set properly while the flavors develop.