Add the olive oil, garlic, onion, potatoes, celery, and bay leaves to a slow cooker set to High heat. Season with salt and pepper, and stir until well combined. Cover and cook, stirring occasionally, until vegetables are softened, about 2 hours.
Season the chicken breasts with salt and pepper, and add to the slow cooker. Set the heat to Low, cover, and cook 1½ hours.
Add the thyme, tomato paste, red wine, and mushrooms. Season well with salt and pepper. Stir until well combined. Leave the heat set to Low, cover, and cook 2 hours, stirring occasionally.
Use a slotted spoon to transfer the chicken and vegetables to a serving platter. Add flour and butter to the sauce that remains in the slow cooker. Whisk until well combined. Pour the sauce through a fine mesh strainer to remove any food particles. Pour the sauce all over the chicken and vegetables and serve immediately.