Heat ¼ inch of oil in a medium skillet over medium-high heat until hot but not smoking. The oil is ready if when you dip the edge of a tortilla into it, it bubbles happily around the edges without going wild. Using tongs, dip the tortillas one at a time in the oil for 10 to 15 seconds per side. The tortillas will ﬁrst appear to soften and then begin to crisp. Drain on paper towels and cover with aluminum foil or an inverted plate to keep warm. (Alternatively, you can lightly toast the tortillas on a well-heated comal or in a skillet over medium heat for 30 seconds per side.)
Pour off all but 2 tablespoons of the oil you used for the tortillas if you fried them. If you toasted the tortillas, add 2 tablespoons oil. Heat the oil over medium heat. Crack 2 eggs into the pan, sprinkle with salt to taste, and cook to your preferred doneness. Keep warm while you repeat with the remaining eggs.
To serve, place 2 of the warmed tortillas on four plates and top with 2 eggs. Ladle a generous amount of green sauce over one of the eggs on each plate and red sauce over the other. Serve with the refried beans on the side and a sprinkling of the cheese.