Place a large, heavy skillet over medium heat. Add the olive oil and heat for 2 minutes. Place the lamb burgers, prosciutto-covered side down in the pan and cook over medium heat until the prosciutto is golden, about 6 to 8 minutes. Turn the burgers and finish cooking, about 6 to 8 minutes more.
Remove the burgers from the pan and place on a serving platter or individual plates. Top each burger with 2 to 3 basil leaves, 1 to 2 slices of tomato, and a drizzle of extra-virgin olive oil and balsamic vinegar. Serve immediately.