Combine gingersnaps and 1/3 cup butter in a bowl; press cookie mixture into a 9-inch baking dish to make a crust.
Mix mashed peaches with flour, brown sugar, cinnamon, and nutmeg in a large bowl until well blended. Gently fold sliced peaches and lemon used into mashed peach mixture. Pour peaches into prepared crust, lightly pressing the peaches down.
Bake in the preheated oven until bubbly, about 40 minutes. Stir white sugar and 1 1/2 tablespoons butter in a small bowl; drizzle butter syrup over peach filling.
Preheat the oven's broiler.
Return pie to the oven and broil until syrup is caramelized, 5 to 10 minutes. Let pie cool before serving.