Preheat the oven to 400 degrees F. Lightly grease the wells of a 12-cup muffin pan with nonstick cooking spray.
In a medium bowl, whisk the butter, oil and sugar together until well combined. Add the eggs one at a time and whisk to combine. In another medium bowl, whisk the flour, cornmeal, baking powder, cayenne and salt to combine.
Add half of the flour mixture to the wet ingredients and stir with a rubber spatula to combine. Add the buttermilk and stir to combine, then add the remaining flour mixture and stir well to incorporate.
Use a 1/4-cup scoop to divide the batter among the wells of the prepared muffin pan, filling each just over 3/4 of the way full. Press 1 cube of cheese into the center of each muffin, submerging it fully without reaching the bottom of the pan.
Bake until the surface of the muffin springs back gently to the touch, 17 to 20 minutes, rotating the pan halfway through baking. Let cool inside the pan for 5 to 10 minutes, then remove from the pan. These muffins are best served immediately, warm, so the cheese inside is still gooey.