How to Make It
In a large skillet, heat the olive oil. Add the scallion whites and garlic and mix well. Cover and cook over low heat, stirring occasionally, until tender, about 5 minutes. Add the baharat and cook, stirring, for 1 minute. Add the tomatoes, season with salt and pepper and bring to a simmer. Cook over moderately low heat, stirring occasionally, until softened and saucy, about 10 minutes. Add the scallion greens and cook until tender, 2 minutes longer. Scrape the mixture into a small bowl and let cool to room temperature.
Heat a large nonstick skillet. Melt 1 1/2 tablespoons of the butter in the olive oil. In a medium bowl, beat the eggs with a fork. Add them to the skillet, season with salt and pepper and cook over moderately low heat, stirring, until curds start to form, 4 to 5 minutes. Stir in the remaining 1 1/2 tablespoons of butter and cook until the eggs are just set, 1 to 2 minutes longer.
Arrange 1 slice of cheese on the bottom of each English muffin. Spoon 1 tablespoon of the scallion-tomato sauce evenly on the cheese and top with half of the eggs. Repeat with the remaining cheese, scallion-tomato sauce and eggs. Close the sandwiches and serve.