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 9×13-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.
Optionally, the baking pan can be lined with pie crust before adding the mixture, but fewer biscuits should be used in preparation or the entire dessert may contain too much dough.
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