Grate cheese into a small bowl. Pull cilantro leaves from stems and chop (a little stem is okay). Curl your cilantro-holding fingers into a “claw” (with fingertips bent underneath your knuckles) to protect them as you chop.
Crack eggs into another small bowl, removing any shell with a spoon. Whisk in yogurt and a pinch each of salt and pepper until completely combined.
Heat 1 tsp. oil in a small frying pan over medium heat. It’s ready when a bead of water sizzles and evaporates immediately. When pan is hot, pour eggs in all at once and turn heat to low. Cook eggs, stirring slowly and gently but constantly, and folding eggs over on themselves, until just set (firm), about 2 minutes. Sprinkle with a pinch each of salt and pepper. Remove from heat.
Drizzle remaining ½ tsp. oil into a large frying pan. Set over medium-high heat. When hot (oil will begin to shimmer), spread over pan with a heatproof spatula.
Meanwhile, rub tortillas with the lime quarter. Using the spatula, lay each tortilla flat in pan and heat until they start to soften, about 1 minute. Flip over with spatula, then sprinkle with cheese. When cheese is melty, move tortillas to a plate. Turn off stove.
Heap eggs on one side of each tortilla. Fold tortillas in half, pressing down gently to close, and top with cilantro; add salsa if you like. Serve warm.