Melt butter in a 10- to 12-inch frying pan over medium heat; stir in brown sugar until dissolved and bubbly. Stir in apples and cook until they are barely tender when pierced, about 5 minutes. Divide the apple-and-sugar mixture equally among 12 buttered muffin cups (1/2-cup capacity).
In a bowl, mix cornmeal, flour, granulated sugar, baking soda, and salt.
In another bowl, mix buttermilk, egg, and oil to blend. Stir into cornmeal mixture just until evenly moistened. Spoon batter equally into muffin cups.
Bake in a 350° regular or convection oven until tops are lightly browned and feel firm when pressed, 15 to 20 minutes. Immediately run a knife between each muffin and cup rim and invert over a baking sheet to remove muffins (if any apple slices remain inside muffin cups, replace on muffins). Serve warm or at room temperature, apple side up.