In a large bowl combine the bread crumbs, parsley, egg, milk, cheese, sun-dried tomatoes, salt, and pepper, Stir to combine. Add the lamb and stir until incorporated. Divide the mixture into 6 (1-inch) thick burgers. Place the slices of prosciutto on a cutting board or piece of parchment paper. Place 1 lamb burger in the center of each slice of prosciutto and wrap the prosciutto around the burger.
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.