Pat chicken dry inside and out and season with salt and pepper, leaving the cavity empty. Melt butter in a skillet over medium heat and cook whole garlic cloves for 5 minutes, turning often without burning them.
Add wine and simmer for 3 minutes. Pour garlic, wine and butter into a 6-quart oval slow cooker, add broth and thyme, and place the chicken breast-side up over the garlic.
Cover and cook on low for 5 to 6 hours until the thickest breast reaches 165°F and the thigh reaches at least 175°F, without touching bone with the thermometer.
Heat the broiler and oil a small roasting pan. Support the bird underneath with two broad spatulas, transfer carefully to the pan and broil 6 inches from the heat for 4 to 6 minutes until the skin is lightly browned.
Rest chicken for 15 minutes. Meanwhile, skim fat from the cooking juices and blend them with the softened garlic until smooth, venting steam carefully.
Simmer the garlic sauce in a saucepan for 5 to 8 minutes until slightly reduced. Carve chicken into 6 servings and spoon the sauce alongside.