Step 1: Preheat your oven to 375°F (190°C). Lightly grease an 8×8 inch baking dish or a similarly sized oven-safe dish.
Step 2: Prepare the peaches. If using fresh peaches, peel (optional), pit, and slice them into 1/2-inch wedges. If using frozen peaches, thaw completely and drain any excess liquid. Place the sliced peaches in a large bowl.
Step 3: In the bowl with the peaches, add 1/4 cup brown sugar, 1 tablespoon all-purpose flour, 1 teaspoon vanilla extract, 1/2 teaspoon ground cinnamon, and a pinch of nutmeg. Gently toss everything together until the peaches are evenly coated. Pour the peach mixture into your prepared baking dish and spread it into an even layer.
Step 4: Prepare the crumble topping. In a separate medium bowl, combine 1/2 cup all-purpose flour, 1/2 cup old-fashioned rolled oats, 1/2 cup packed light brown sugar, and 1/4 teaspoon ground cinnamon. Stir these dry ingredients together.
Step 5: Add 1/4 cup (1/2 stick) cold unsalted butter, cut into small cubes, to the dry crumble mixture. Using your fingertips or a pastry blender, work the butter into the dry ingredients until the mixture resembles coarse crumbs with some larger pea-sized pieces.
Step 6: Evenly sprinkle the crumble topping over the peach mixture in the baking dish.
Step 7: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbling around the edges. If the topping starts to brown too quickly, loosely tent the dish with aluminum foil.
Step 8: Remove from the oven and let it cool for at least 10-15 minutes before serving. This allows the filling to set slightly. Serve warm.