In a large bowl, mix the turkey, mushrooms, kale, onion, oats, balsamic vinegar, 1 tsp. salt and 3/4 tsp. pepper until just blended. Form into 8 balls (about a rounded 1/4 cup each).
Coat a large griddle or nonstick skillet with cooking spray; heat over medium.
Working in batches if needed, add the balls to the skillet. Using your fingers, gently flatten into 3-inch patties. Cook, turning once, until browned and cooked through, about 5 minutes per side. During the last 2 minutes of cooking, top each patty with a piece of cheese.
Divide the patties among the roll bottoms. Add toppings and roll tops.