Melt butter in 12-inch skillet over medium-high heat. Add onion, carrot, celery and garlic. Cook until the onion is softened and lightly browned, about 10 minutes. Stir in flour and cook until golden brown, about 2 minutes. Stir in 1 cup broth, scraping up any browned bits and smoothing out any lumps; transfer to slow cooker.
Stir in remaining 2 cups broth, wine, thyme sprigs and bay leaves into the slow cooker. Season the turkey breast with salt and pepper and place, skin side up, into slow cooker. Cover and cook until turkey reaches 165 degrees on instant-read thermometer, 5 to 7 hours on low.
Transfer turkey to cutting board, tent loosely with aluminum foil, and let rest for 20 minutes. Let braising liquid settle for 5 minutes, then remove fat from surface using a large spoon. Strain braising liquid into saucepan, discarding solids, and simmer until thickened, about 15 minutes, adding more flour diluted in water, if necessary. Season with salt and pepper to taste.
Carve turkey and serve with gravy.