Heat oil in a medium-sized skillet on medium heat. Add the chopped tomatillos, onion, jalapeño pepper, and a small squeeze of lime juice. Cook on medium to medium-low heat (you want to gently cook, not brown) for about 5 minutes, stirring occasionally, until the onions have softened, and the tomatillos are no longer bright green.
Add the eggs directly to the pan (no need to whisk first). Break up the yolks with your stirring spoon. Cook gently, stirring and scraping the bottom of the pan with a wooden spoon. Salt and pepper to taste. Remove from heat when the eggs begin to set, but are still moist, about 3 minutes. Serve immediately. Sprinkle on cilantro for garnish and serve with a dish of lime wedges and hot sauce on the table.