Preheat oven to 425 degrees, with racks in upper and lower thirds.
Toss garlic, bread, cheese, and tomatoes with 3 tablespoons oil; season with salt and pepper.
Arrange in a single layer on a rimmed baking sheet.
Rub chicken with remaining 2 tablespoons oil; season with salt and pepper.
Place on bread mixture, skin-sides up, along with thyme.
Roast in lower third, flipping chicken and bread mixture once, 25 minutes.
Meanwhile, remove ribs from kale; tear into 2-inch pieces.
Drizzle with oil and season with salt and pepper.
Switch oven to broil; remove thyme from baking sheet.
Transfer chicken to a plate.
Scatter kale over bread mixture; arrange chicken on top.
Broil until chicken skin is golden and crisp, 2 to 4 minutes.
Transfer chicken to serving plates; toss bread mixture and serve, with more cheese.
