Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the steak with ½ teaspoon salt and ¼ teaspoon pepper.
Cook the steak, 4 to 5 minutes per side for medium-rare. Remove from skillet and let rest for 5 minutes before slicing.
Add the tomatoes to the skillet and cook, cut-side down, until browned, 2 to 3 minutes.
Meanwhile, heat the remaining teaspoon of oil in a nonstick skillet over medium heat.
Crack the eggs into the nonstick skillet and cook, covered, to the desired doneness, 2 to 4 minutes for slightly runny sunny-side-up eggs.
Sprinkle the tomatoes and eggs with the oregano and 1/8 teaspoon each salt and pepper. Serve with the steak.