Step 1: Preheat your oven to 375°F (190°C). Lightly grease an 8x8 inch baking dish or a similar-sized oven-safe dish.
Step 2: In a large bowl, combine the sliced apples, 1/4 cup granulated sugar, 1/4 cup brown sugar, 1 tsp cinnamon, nutmeg, lemon juice, and 2 tablespoons of flour. Toss gently until the apples are evenly coated. Pour the apple mixture into the prepared baking dish.
Step 3: To make the crumble topping, in a separate medium bowl, combine 1 cup all-purpose flour, rolled oats, 1/2 cup brown sugar, 1/4 cup granulated sugar, and salt. Mix well.
Step 4: Add the cold, cubed butter to the dry ingredients. Using your fingertips, a pastry blender, or a food processor, work the butter into the flour mixture until coarse crumbs form. Be careful not to overmix; you want a crumbly texture.
Step 5: Evenly sprinkle the crumble topping over the apple mixture in the baking dish.
Step 6: Bake for 35-45 minutes, or until the topping is golden brown and the apple filling is bubbling and tender when pierced with a knife. If the topping starts to brown too quickly, you can loosely tent it with aluminum foil.
Step 7: Remove from the oven and let it cool for at least 10-15 minutes before serving. This allows the filling to set slightly.