For the gremolata: Heat a large skillet over medium heat. Add the breadcrumbs and olive oil and toast, stirring often with a wooden spoon until golden brown. Place in a small bowl and while still hot, add the lemon zest, red pepper flakes and salt. Mix together and cool the mixture to room temperature. Stir in the parsley and cheese.
For the asparagus: Wipe the pan with a clean dish towel. Place over medium-high heat and add 2 tablespoons of the olive oil. Add the asparagus and 1/4 teaspoon of the salt. Cook, stirring and flipping often with tongs until the asparagus is bright green, cooked through and slightly browned, 4 to 5 minutes.
Divide the asparagus among four serving plates. To the same pan, add the remaining 2 tablespoons of olive oil and place over medium-high heat. Crack 2 of the eggs into the pan, being careful as it may splatter, sprinkle with salt and reduce the heat slightly and cook until the edges are light brown and crispy and the whites are just set, 2 to 3 minutes. The yolk should still be runny. Repeat for the remaining 2 eggs. Using a slotted spatula, slide an egg on each plate. Sprinkle each dish with the breadcrumb gremolata and a drizzle of olive oil if desired.