Finely chop enough onion to yield 1/3 cup (reserve remaining rounds for burgers). In a large skillet, heat oil over medium-high; cook chopped onion and mushrooms, stirring occasionally, until onion is soft and mushrooms are golden, about 4 minutes. Season with salt and pepper, then transfer to a cutting board and roughly chop mixture.
Heat a grill to medium-high. Clean and lightly oil hot grill. Season onion rounds with salt and pepper; grill until lightly charred and tender, 5 minutes, flipping once. In a large bowl, combine mushroom mixture, turkey, and parsley. Form into four 1/2-inch-thick patties; season with salt and pepper. Grill, about 4 minutes, flipping once. Top each patty with cheese; grill until burgers are cooked through and cheese is just melted, 1 minute. Serve burgers on buns with grilled onions, peppers, and arugula.