Preheat oven to 350°F (175°C).

Mix peaches, white sugar, brown sugar, butter, cornstarch, vanilla extract, ground cinnamon, ground 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 cinnamon, white sugar, and nutmeg. Flatten biscuits on a lightly floured surface using a rolling pin. Cut each flattened biscuit into 12 to 15 small squares and 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.