Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: Peel, core, and slice the apples into 1/4-inch thick pieces. Place them in a large bowl.
Step 3: In a small bowl, whisk together 1/2 cup granulated sugar, 1/4 cup flour, 1 teaspoon cinnamon, and 1/4 teaspoon nutmeg. Pour this mixture over the sliced apples, add the lemon juice, and toss gently until the apples are evenly coated.
Step 4: Pour the apple mixture into the prepared baking dish and spread evenly.
Step 5: For the topping, in a separate medium bowl, combine 1 cup flour, brown sugar, 1/2 cup granulated sugar, 1 teaspoon cinnamon, 1/2 teaspoon nutmeg, salt, and rolled oats. Mix well.
Step 6: Add the cold, cubed butter to the dry topping ingredients. Use your fingers, a pastry blender, or a fork to cut the butter into the mixture until it resembles coarse crumbs with some larger pea-sized pieces.
Step 7: Sprinkle the crumble topping evenly over the apple mixture in the baking dish.
Step 8: Bake for 45-55 minutes, or until the topping is golden brown and crispy, and the apples are tender and bubbly around the edges. If the topping browns too quickly, you can loosely tent it with foil.
Step 9: Remove from the oven and let cool for at least 15-20 minutes before serving. This allows the filling to set slightly. Serve warm with vanilla ice cream or whipped cream, if desired.