Crack egg into a large bowl. Add milk, Pecorino Romano, salt and pepper; whisk to combine. Stir in sun-dried tomatoes, parsley and bread crumbs. Add lamb and, using your fingers, combine just until mixed. (Take care not to over-mix or your burger's texture will be tough.)
Place prosciutto slices onto plate or cutting board side by side, a few inches apart.
Divide lamb mixture into 4 equal pieces and shape into 4, 1-inch thick burgers. Place one burger on the center of a prosciutto slice and wrap with prosciutto. Repeat with remaining burgers.
Heat olive oil in a large skillet over medium heat until hot, about 2 minutes. Place lamb burgers, prosciutto-covered side down in pan and cook until prosciutto is golden, 6 to 8 minutes. Turn burgers and cook, 6 to 8 minutes more.
Transfer burgers to serving platter or to individual plates. Top each burger with 2 basil leaves and 1 slice of tomato; drizzle with balsamic vinegar and serve.