Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9-inch square baking dish or an 8x8 inch oven-safe dish.
Step 2: Prepare the peaches. Peel, pit, and slice your peaches into roughly 1/2-inch thick wedges. In a large bowl, gently combine the sliced peaches with granulated sugar, all-purpose flour, vanilla extract, ground cinnamon, ground nutmeg, and fresh lemon juice. Toss until the peaches are evenly coated.
Step 3: Pour the peach mixture into your prepared baking dish, spreading them out evenly.
Step 4: Prepare the crumble topping. In a separate medium bowl, combine the old-fashioned rolled oats, all-purpose flour, light brown sugar, granulated sugar, ground cinnamon, ground nutmeg, and salt. Whisk everything together until well combined.
Step 5: Add the cold, cubed unsalted butter to the dry crumble ingredients. Using your fingertips, a pastry blender, or two knives, cut the butter into the mixture until coarse crumbs form. You want some pea-sized pieces of butter remaining; these will melt and create pockets of delicious crispiness in your topping. Don't overmix!
Step 6: Evenly sprinkle the crumble topping over the peach filling in the baking dish. Make sure the peaches are mostly covered to create a uniform, golden crust.
Step 7: Bake for 35-40 minutes, or until the topping is golden brown and the peach filling is bubbly around the edges. If the topping starts to brown 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 and prevents it from being too runny. Serve warm, perhaps with a scoop of vanilla bean ice cream or a dollop of fresh whipped cream.