For the cornbread: Preheat oven to 425 degrees. Butter a 9-inch square baking pan. In a medium bowl, combine all the cornbread ingredients until just moistened. Pour into the pan and bake for 20 to 25 minutes, or until the bread is cracked on top, brown on the edges and pulling away from the pan. Let cool in pan for 10 minutes, then turn out onto a wire rack. When completely cool, slice into 1/2-inch slabs.
For the sandwiches: Using the broiler, toast cornbread slabs until lightly browned. Spread one side of each slab with butter, then sprinkle the cheese over two of the buttered slabs. Return the cheesy slabs to broiler until cheese begins to melt, about one minute. Remove immediately.
Over high heat, melt the 2 tablespoons of butter in a sauté pan. Fry the eggs sunny side up until the edges are browned and the yolks are set, but still yellow.Place a cheesy cornbread slab on each of the two plates. Top each with 2 eggs, some cilantro leaves, hot sauce and extra cheese if desired. Season to taste with salt and pepper. Top with remaining cornbread slabs.