Preheat the oven to 350 degrees F (175 degrees C).

Combine butter and 1 1/4 cups sugar in a 9-inch oven-safe skillet over medium heat. Cook and stir until sugar dissolves and mixture begins to caramelize, 5 to 10 minutes. When mixture starts to simmer, add diced apples and sprinkle with cinnamon and nutmeg. Gently press on the apples with a spatula until coated with sugar mixture and simmer, without stirring, until they begin to soften, about 5 minutes.

Combine baking mix, milk, shredded Cheddar cheese, and remaining 1/4 cup sugar in a bowl and stir until consistency is similar to a waffle batter. Spoon mixture evenly over the apples in the skillet; no need to be too precise as it will spread out while baking.

Bake skillet in preheated oven until batter reaches a firm consistency and is golden brown, 20 to 25 minutes. Remove from the oven and allow to cool on the stovetop until the caramelized sugar thickens a bit, about 5 minutes. Spoon servings onto individual plates, drizzle with caramel liquid, and serve while warm.