Heat 1 tbsp. oil in a large frying pan over high heat. Add asparagus and sauté until tender when pierced, about 5 minutes. Transfer asparagus to a plate. Add remaining oil and rosemary to pan. Cook rosemary until oil is hot, about 2 minutes, then discard the herb.
Put bread slices in pan and toast in oil, turning once, about 1 minute. Sprinkle about 2 tbsp. parmesan onto each piece of bread and cook until cheese starts to melt and bread starts to brown slightly, about 2 minutes. Transfer bread to plates.
Reduce heat to medium and in the same pan, gently fry both eggs to your liking. Top each piece of bread with 2 prosciutto slices and half the asparagus. Set an egg on top of each and sprinkle servings with salt and pepper.