Preheat oven to 350°F. Heat 1 Tbsp. oil in medium saucepan over medium heat. Add onion and cook, stirring occasionally, until softened, about 5 minutes. Add pinto beans and mash coarsely with a potato masher or a fork. Remove from heat and stir in cheese. Keep warm.
Meanwhile, heat 1/2 Tbsp. oil in a nonstick, 12-inch skillet over medium heat. Add 1 tortilla and cook for 1 minute per side until heated through and pliable. Transfer to a foil-lined baking sheet. Spread one quarter of bean mixture on top of tortilla; keep warm in a 250°F oven. Repeat with remaining tortillas and bean mixture, adding another 1/2 Tbsp. oil to skillet as needed.
Add 1/2 Tbsp. oil to skillet and then add 4 eggs. Cook to your liking. Transfer 2 eggs onto each of 2 tortillas; season with salt and pepper and sprinkle diced tomato on top. Cook and serve 4 more eggs in same manner. Serve tortillas with salsa.