Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9×13 inch baking dish.
Step 2: In a large bowl, combine the sliced peaches, 1/2 cup granulated sugar, 1/4 cup light brown sugar, 2 tablespoons all-purpose flour, 1 teaspoon ground cinnamon, 1/4 teaspoon ground nutmeg, and 1 tablespoon lemon juice. Toss gently until the peaches are evenly coated.
Step 3: Pour the peach mixture into the prepared baking dish, spreading it evenly.
Step 4: In a separate medium bowl, combine 1 1/2 cups all-purpose flour, the remaining 1/4 cup granulated sugar, the remaining 1/2 cup light brown sugar, 1 teaspoon ground cinnamon, and 1/4 teaspoon salt. Whisk these dry ingredients together.
Step 5: Add the 1/2 cup (1 stick) cold, cubed unsalted butter to the dry ingredients for the topping. Use your fingertips, a pastry blender, or a fork to cut the butter into the flour mixture until it resembles coarse crumbs.
Step 6: Sprinkle the crumble topping evenly over the peach filling in the baking dish.
Step 7: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbly around the edges and the fruit is tender. If the topping browns too quickly, you can lightly tent it with aluminum foil.
Step 8: Remove from the oven and let it 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.