Preheat oven to 350 degrees F (175 degrees C).

Mix peaches, 3/4 cup white sugar, brown sugar, butter, cornstarch, vanilla extract, 1 teaspoon cinnamon, 1/2 teaspoon nutmeg, and salt in a large pot over medium heat; cook and stir until sugars are dissolved and butter is melted, about 5 minutes.

Sprinkle each biscuit with 1 pinch cinnamon, 1 pinch white sugar, and 1 pinch nutmeg. Flatten biscuits on a lightly floured surface using a rolling pin. Cut each flattened biscuit into 12 to 15 small squares; stir into peach mixture. Reduce heat to medium-low and simmer, stirring occasionally, until thickened, about 10 minutes.

Pour peach-biscuit mixture into a 9x13-inch baking dish and allow to thicken for 2 to 3 minutes; cover or decorate with pie crust. Cover dish with a sheet of aluminum foil and place dish on a baking sheet.

Bake in the preheated oven until crust is golden brown, 45 minutes to 1 hour. Cool before serving.