Heat 2 tsp. olive oil in nonstick skillet over medium high. Cook mushrooms, stirring, 4 minutes. Add spinach, thyme, 1/2 tsp. salt, and 1/4 tsp. pepper. Cook for 1 minute. Transfer to a bowl.
Toss turkey with spinach mixture and form into 6 patties.
Add remaining 2 tsp. olive oil to skillet and cook onions over medium heat with remaining 1/4 tsp. salt and 1/4 tsp. pepper, stirring occasionally, until golden brown, about 10 minutes. Transfer to a bowl.
Lightly coat patties, cut sides of buns, and the grill rack (or nonstick skillet) with cooking spray. Over medium heat, cook burgers, turning, until golden brown and cooked through, about 5 minutes per side. Top with cheese in last minute of cooking and let melt. Toast buns on grill for 2 minutes.
Place patties on the buns and top with caramelized onions.
