Preheat a panini press. Lightly brush the bread on both sides with extra-virgin olive oil. Wrap each asparagus spear with a slice of prosciutto. Place 1 slice of cheese on one slice of bread, then the smoked turkey. Top with the prosciutto-wrapped asparagus and the remaining 1 slice of cheese and 1 bread slice. Grill until the bread is crisp and cheese is melted.