Prepare the apple filling: In your microwave-safe mug, combine the diced apple, brown sugar, cinnamon, nutmeg, and lemon juice. Stir well to coat all apple pieces evenly with the sugar and spices.
Microwave the filling: Cover the mug with a paper towel and microwave on high for 1 minute. This partially cooks the apples and helps them release their juices.
Make the crumble topping: In a separate small bowl, mix together the flour, oats, granulated sugar, and salt. Add the cold butter pieces and use a fork to cut the butter into the dry ingredients until the mixture resembles coarse crumbs.
Add the topping: Remove the mug from the microwave and stir the apple mixture. Then sprinkle the crumble topping evenly over the apples, pressing down very lightly.
Final cooking: Microwave uncovered for an additional 1-2 minutes until the topping looks set and the apple filling is bubbling at the edges. Microwaves vary in power, so start with 1 minute and check before adding more time.
Cool slightly: Allow the crumble to cool for 1-2 minutes—the mug and contents will be extremely hot, and the filling will thicken slightly as it cools.
Serve: Top with a small scoop of vanilla ice cream or a dollop of whipped cream if desired, and enjoy while still warm!