Step 1: Preheat your oven to 375°F (190°C). Lightly grease an 8x8 or 9x9 inch baking dish.
Step 2: In a large bowl, combine the peeled, cored, and sliced apples with 1/2 cup granulated sugar, 1/4 cup packed light brown sugar, 1 tablespoon flour, 1 teaspoon cinnamon, 1/4 teaspoon nutmeg, and 1 tablespoon lemon juice. Toss gently to ensure all apple slices are coated.
Step 3: Pour the apple mixture into the prepared baking dish. Dot the top of the apples with the 2 tablespoons of butter, cut into small pieces.
Step 4: In a separate medium bowl, prepare the crumble topping. Whisk together 1 cup all-purpose flour, 1/2 cup packed light brown sugar, 1/2 cup granulated sugar, 1/2 teaspoon cinnamon, and a pinch of salt.
Step 5: Add the cold, cubed butter to the dry topping ingredients. Use your fingertips or a pastry blender to cut the butter into the mixture until it resembles coarse crumbs with some pea-sized pieces of butter remaining. If using, stir in the 1/2 cup rolled oats.
Step 6: Evenly sprinkle the crumble topping over the apple mixture in the baking dish. Do not press it down, as this will prevent it from becoming crispy.
Step 7: Bake for 40-50 minutes, or until the topping is golden brown and the apple filling is bubbling and tender when pierced with a knife. Baking time may vary depending on your oven and the apple variety.
Step 8: Remove the crumble from the oven and let it cool for at least 15-20 minutes before serving. This allows the filling to set and prevents it from being too runny.