In 10-inch non-stick skillet over medium heat, in 1 Tbsp. hot olive oil, cook onions 7 to 8 minutes or until browned and almost tender, stirring occasionally. Add vinegar and 1/8 tsp. salt. Reduce heat to low; cover and cook 3 to 5 minutes or until tender; set aside.
In 10-inch skillet over medium-high heat, in remaining 1 Tbsp. hot olive oil, cook mushrooms 2 minutes, stirring frequently. Add garlic, thyme and lemon zest and cook 2 minutes or until mushrooms are tender; cool.
In bowl, mix ground veal, mushroom mixture and remaining 1/4 tsp. salt. Shape into four 3/4-inch-thick patties.
Prepare a charcoal or gas grill or use a grill pan on top of stove over medium heat. Place rolls, cut-side down on grill; lightly toast. Remove rolls. Grill burgers 10 to 14 minutes, turning once or until internal temperature of 160° F. Place roll bottoms on serving platter. Top each bottom with lettuce leaf. Place burger on top of lettuce. Top each burger with sliced cheese, some balsamic onions and roll tops.