Step 1: Preheat your oven to 375°F (190°C). Lightly butter an 8×8 inch (or similar 2-quart) baking dish.
Step 2: In a large bowl, combine the prepared plum pieces with 1/4 cup granulated sugar, 2 tablespoons light brown sugar, 1 tablespoon cornstarch, 1 teaspoon vanilla extract, 1/2 teaspoon ground cinnamon, and 1/4 teaspoon ground nutmeg. Mix gently until the plums are evenly coated. Pour the plum mixture into your prepared baking dish, spreading it out evenly.
Step 3: In a separate medium bowl, combine 1 cup rolled oats, 1/2 cup all-purpose flour, 1/2 cup light brown sugar, 1/4 teaspoon ground cinnamon, and a pinch of salt. Whisk these dry ingredients together thoroughly.
Step 4: Add 1/2 cup (1 stick) cold, cubed unsalted butter to the dry ingredients. Using your fingertips, a pastry blender, or two knives, cut the butter into the mixture until it resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 5: Sprinkle the oat crumble topping evenly over the plum filling in the baking dish, ensuring the plums are covered.
Step 6: Place the baking dish on a baking sheet (to catch any potential overflow) and bake for 35-45 minutes, or until the topping is golden brown and the plum filling is bubbling around the edges. If the topping browns too quickly, loosely tent with aluminum foil.
Step 7: Once baked, remove the crumble from the oven and let it cool on a wire rack for at least 15-20 minutes before serving. This allows the fruit juices to set.
Step 8: Serve warm, optionally with vanilla ice cream, Greek yogurt, or crème fraîche.