Step 1Heat dry heavy skillet or griddle over medium heat. Tear ancho and guajillo chiles into flat pieces, then toast a few at a time: Use metal spatula to press chile pieces flat against hot surface, skin side up, until aromatic and lightened in color underneath, about 10 seconds. (If heat is right, you'll hear a slight crackle when you press down, but you shouldn't see more than slightest wisp of smoke.) Place in bowl, cover with hot water, lay small plate on chiles to keep submerged and soak 20 minutes to rehydrate.
Step 2Use tongs to transfer rehydrated chiles to food processor or blender. Measure 1 1/2 cups soaking liquid and add to chiles along with garlic. Blend to smooth puree, then push through medium-mesh strainer into pie plate. Consistency should be like that of canned tomato sauce. Taste (sauce will be little rough on tongue at this point) and season highly with salt, usually about 1/2 teaspoon. Set aside 1/4 cup sauce for potatoes.
Step 1Simmer potatoes in salted water to cover until just barely tender, about 10 minutes. Drain.
Step 2In large (12-inch) skillet (preferably nonstick), heat oil or lard over medium heat. Add onion and cook, stirring regularly, until translucent, about 5 minutes. Add potatoes and continue cooking, turning and scraping up any browned bits, until richly browned and crusty, 15 to 20 minutes. Stir in 1/4 cup sauce and cook another 5 minutes, then remove from heat.
Step 1While potatoes are cooking, make enchiladas. Place sauce and tortillas near stove with baking sheet beside them. Set small (8-inch) skillet, preferably nonstick, over medium heat and add 1 tablespoon oil or lard. When oil is very hot (just beginning to smoke), dip both sides of 1 tortilla into sauce, then lay in oil. Let sear and sizzle about 20 seconds, then use small spatula to flip it over. Sear other side another 20 seconds. Transfer to baking sheet, laying flat. Adding oil to skillet as needed, continue dipping and frying remaining tortillas, wiping out skillet occasionally. Lay tortillas on baking sheet in groups of two, about half overlapping each other.
Step 2Divide potatoes among chile-seared tortillas and place in 350-degree oven to warm, 5 to 10 minutes. Meanwhile, wash and dry large skillet and set over medium to medium-low heat. Add 1 tablespoon oil or lard. When warm (not too hot), crack in eggs, cover pan and cook until eggs are as done as you like, about 3 to 4 minutes for set whites and slightly runny yolks.
Step 3While eggs are cooking, toss lettuce in small bowl with vinegar and a good sprinkling of salt. Put an enchilada-potato stack on each of four dinner plates and top with 1 fried egg, then a portion of dressed lettuce and generous sprinkling of grated cheese.