How to Make It
Preheat oven to 300°F. Place enchilada sauce in a small saucepan and cook over medium, stirring occasionally, until hot, about 5 minutes. Reduce heat to low and simmer to keep warm.
Meanwhile, heat oil in a medium nonstick or cast-iron skillet over medium-high. Cook tortillas, one at a time, until just starting to change color, but are still limp and pliable, about 30 seconds per side. Transfer to a paper towel-lined plate to drain excess oil. Pour all but 2 tablespoons oil from skillet to a heat-proof bowl; discard when cool. Reserve 2 tablespoons oil in skillet off heat.
Dip one tortilla in warm enchilada sauce, so that sauce lightly coats both sides. Place tortilla on an oven-safe plate or baking sheet. Spoon about 2 tablespoons sauce over tortilla, then sprinkle with about 3 tablespoons cheese. Dip a second tortilla in warm enchilada sauce, so that sauce lightly coats both sides. Layer tortilla on top of first tortilla on plate or baking sheet. Spoon another 2 tablespoons sauce over tortilla, then sprinkle with another 3 tablespoons cheese. Repeat process with remaining tortillas, sauce, and cheese so that you have 4 total enchilada stacks. Bake in preheated oven until cheese is melted, about 7 to 9 minutes.
While enchiladas are baking, heat reserved 2 tablespoons oil in skillet over medium. Add 2 eggs to oil in skillet; season with half of salt. Cook eggs until desired degree of doneness, about 2 to 3 minutes per side for over-easy. Transfer to a plate once cooked; set aside. Repeat with remaining eggs and salt.
When cheese is melted, remove enchiladas from oven. Top each stack with one egg and garnish evenly with cilantro, tomato, onion, and lettuce.