Fill a saucepan with 1 inch of water. Bring water to a low boil. Carefully add eggs and cover with lid. Boil 6 minutes.
While eggs are boiling, wrap pieces of prosciutto around each asparagus spear until covered. Set aside.
Remove eggs from water and immediately place them into an ice bath for 5 minutes.
While eggs are cooling down, heat olive oil in a large pan over medium heat. Place prosciutto-wrapped asparagus on pan and cook 3 minutes. Turn asparagus over and cook 3 minutes longer.
Peel the top of each egg and slice open with a spoon to expose the soft yolk. Serve asparagus spears with soft boiled eggs.