Spread half the softened butter on the cut sides of the corn muffin halves. In a medium skillet, cook the muffins cut side down over medium heat until toasted, about 3 minutes. Transfer to a plate.
Add the EVOO, 1 turn of the pan, to the skillet. Increase the heat to medium-high, add the sausage and cook, stirring occasionally, until crisp at the edges, 2 to 3 minutes. Add the onion, bell pepper, celery and garlic and season with salt and pepper. Cook, tossing occasionally, until the vegetables are softened, about 5 minutes. Crumble in the muffins. Add the chicken broth and heat through, 1 to 2 minutes. Top the stuffing with the cheese, cover with foil and remove from the heat.
In a medium nonstick skillet, melt the remaining 1-1/2 tablespoons butter over medium heat. In a medium bowl, beat the eggs with the hot sauce and season with salt and pepper. Add the eggs to the skillet and cook, scrambling, until set, about 5 minutes.
Serve a pile of stuffing topped with the scrambled eggs. Sprinkle the scallions on top.