For the filling: Preheat the oven to 375 degrees F.
Combine the apples, sugars, flour, lemon juice, cinnamon, cornstarch, salt and nutmeg in a medium bowl and stir to combine. Melt the butter in a 12-inch skillet over medium heat. Add the apple mixture and cook, stirring frequently, until the apples start to soften, 5 to 8 minutes. Turn off the heat and set aside.
For the streusel topping: Add the flour, oats, brown sugar, cinnamon, salt and butter to a food processor and pulse a couple of times until it resembles coarse, wet sand. (Do not overmix.) Set aside. Coat the insides of 4 oven-safe coffee mugs with cooking spray, then place them in a 9-inch square baking pan. Fill the mugs about three-quarters of the way with the apple mixture, then top each with 1/4 cup streusel topping. Bake until the top is golden brown, 25 to 30 minutes. Cool until warm to the touch, about 20 minutes.