Place garlic and broth in a 6-quart slow cooker. Toss wings with salt and pepper and arrange them over the garlic.
Cover and cook on low for 3 to 4 hours until wings reach 165°F and are tender but still hold together.
Heat the broiler and oil a foil-lined rimmed sheet pan. Transfer wings carefully to the pan and broil 6 inches from the heat for 4 minutes, turn and broil for 3 to 4 minutes more until lightly browned.
Lift the cooked garlic from the broth and mash smooth in a small bowl. Melt butter in a saucepan and stir in the garlic paste for 1 minute.
Brush hot wings with garlic butter and divide Parmesan evenly over them, turning gently to coat. Serve as 6 appetizer portions.