Grated Parmesan becomes irresistibly crisp and lacy when added to a hot skillet, taking ordinary fried eggs to new heights.
Heat a small nonstick skillet over medium until a shred of cheese dropped in begins to sizzle. Spread 1/4 cup cheese in an even layer over bottom of skillet. Heat without disturbing until cheese begins to bubble, 30 seconds to 1 minute. Crack 1 egg directly on top of cheese. Cook until edges are just set, about 2 minutes. Cover and continue cooking until white is set but yolk is still runny, about 1 minute, 30 seconds more. Season with salt and pepper. Run a spatula around edge of egg and cheese to loosen. Transfer to a plate with spatula. Repeat with remaining 1/4 cup cheese and egg.
Top eggs with greens. Squeeze lemon over top and drizzle with oil; season with salt and pepper and serve.