Slice the scallions thin, separating the white and the green parts.
In a large skillet over low heat, heat the olive oil.. Add the white part of the scallions and the garlic. Stir and then cover and cook over low heat for about 5 minutes, until tender. Add the smoked paprika and stir for about 1 minute.
Add the tomatoes, and bring to a simmer. Cook over medium-low heat, stirring occasionally until the tomatoes are soft and the mixture looks "saucy" about 10 minutes. Add the scallion green parts and cook those until tender, about 2 more minutes.
Transfer the mixture to a bowl and let cool to room temperature.
In a large non-stick skillet, heat the olive oil then add the butter to melt. In a bowl, beat the eggs with a fork and add them to the skillet. Season with salt and pepper, and cook over medium-low heat until they start to harden, about 4-5 minutes. Cook until the eggs are just set, about 1 minute longer.
Place 1 slice of cheese on the bottom of each English muffin. Top each piece of cheese with 1 tablespoon of sauce, then half of the eggs. Repeat with the remaining cheese, sauce and eggs. Replace the muffin tops and serve with any remaining sauce.