Place the asparagus on a large cutting board and line up the ends with your hands, then use a chef’s knife to cut about 1 to 2 inches off the tough bottoms so only tender stalks remain.​

Rinse the trimmed asparagus under cool running water in the sink, then dry the spears completely with a clean kitchen towel or paper towels so they sauté well in the skillet.​

Set a large skillet on the stove burner and turn the heat to medium, then add 1 tbsp olive oil and 1 tbsp butter to the pan and move them around with a wooden spoon until the butter melts and mixes with the oil.​

Lay the dry asparagus spears in the skillet in a single layer using tongs so each spear touches the pan, then sprinkle ½ tsp salt and ¼ tsp black pepper over the asparagus with a measuring spoon.​

Cook for 7 minutes over medium heat, turning the spears every minute with tongs so they stay bright green, pick up light golden spots, and turn crisp-tender when pierced with a fork.​

When the asparagus is done, use tongs to move it to two serving plates, dividing the spears evenly, and keep the plates near the stove so they stay warm.​

Crack 2 eggs into a small bowl, one at a time, so you can check for shell pieces, using your hands to remove any bits if needed.​

Place the same skillet back on the burner over medium heat, add 1 tsp olive oil and 1 tbsp butter, and swirl the pan with the wooden spoon until the butter melts and coats the bottom.​

Gently slide the eggs from the bowl into the skillet, then cook for 3 minutes over medium heat until the whites are set and opaque and the yolks stay soft and runny.​

Use a spatula to lift one fried egg from the skillet and place it on top of the asparagus on the first plate, then repeat with the second egg for the second plate.​

Hold a Microplane or fine grater over each plate and grate Parmesan cheese directly on top of the eggs and asparagus until you build a light, even layer that covers the yolks and some of the spears.​

Finish by adding a small pinch of extra black pepper over each egg with your fingers or a measuring spoon, then serve right away while the yolks are still runny.