Brush both sides of each slice of bread with olive oil, and grill, on one side about 30 seconds; flip and repeat.
Remove from grill and set each slice on a serving plate.Grill the whole prosciutto slices until crisp and well colored, about 2 minutes per side.
Lay one slice on top of each piece of bread.
Brush the avocado halves (pitted and peeled) with fresh lime juice and olive oil. Grill, cut side down, for 1 to 2 minutes, creating grill marks. Turn to create crosshatch marks, and grill for 1 to 2 minutes more. Top bread and prosciutto with the grilled avocado.