Heat 1 tablespoon butter in a microwave-safe mug until softened, about 10 seconds. Stir white sugar into butter until dissolved. Mix 1/4 cup flour, applesauce, baking powder, vanilla extract, and salt into butter-sugar mixture until batter is smooth.
Mix 2 teaspoons flour, 1 teaspoon butter, brown sugar, and cinnamon together in a bowl using your hands until the texture of lumpy sand is reached; sprinkle over batter.
Cook in the microwave until coffeecake appears to be barely set, 50 to 80 seconds. Cool before eating.