Fill a large bowl with cold water. Bring a large pot of lightly salted water to a boil. Add asparagus and cook uncovered until blanched, about 2 minutes. Drain in a colander and immediately refresh in cold water for several minutes to stop the cooking process. Drain.
Place 2 asparagus spears side by side and top with 2 half-slices of mozzarella cheese. Top cheese with 2 more asparagus spears. Wrap bundle in a slice of prosciutto so mozzarella cheese is tightly encased. Repeat with remaining asparagus, mozzarella cheese, and prosciutto to create 4 parcels.
Heat oil in a skillet over medium heat. Carefully fry parcels in the hot oil until prosciutto has crisped and mozzarella cheese oozes, about 5 minutes per side. Season with salt and pepper. Serve warm.