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Lightly toast the muffin or biscuit in a toaster oven until just barely browned.
Heat the butter and oil in a large cast iron or nonstick skillet over medium low heat until the butter is melted. When the butter starts to sizzle, crack the egg into the skillet. Cook for 1 minute, letting the white firm up before shifting slightly to the side.
Add the toasted biscuit halves, cut side down, to the skillet. Cook the halves for 20-30 seconds, just long enough to sop up some of the butter. Transfer to a plate and flip the egg back into the center of the skillet.
Place one slice of cheese over the flipped egg and the other slice on one of the biscuit halves. Cook the egg until done to your liking (undercook a bit if you’ll be reheating later) and slide on top of the cheese-covered biscuit half so the egg is layered between both slices of cheese.
Season with salt and pepper if desired before serving. Wrap in waxed paper and foil to store; to reheat, microwave in the waxed paper for 20 seconds on regular power or bake in a toaster oven at 350° in the foil for 5 minutes.