6 (12-ounce) bone-in, skin-on split chicken breasts, trimmed
Salt and pepper
¼ cup vegetable oil
1¼ pounds cremini mushrooms, trimmed and halved if small or quartered if large
2 onions, minced
2 teaspoons minced fresh thyme or ½ teaspoon dried
4 garlic cloves, minced
¼ cup all-purpose flour
¾ cup dry white wine
1 cup low-sodium chicken broth
1 pound carrots, peeled, halved lengthwise and sliced ½ inch thick
2 tablespoons soy sauce
2 bay leaves
½ cup heavy cream
¼ cup minced fresh tarragon
Dry chicken with paper towels and season with salt and pepper. Heat 1 tablespoon oil in 12-inch skillet over medium-high heat until just smoking. Add half of chicken, skin side down, and brown lightly, about 5 minutes; transfer to plate. Repeat with 1 tablespoon more oil and remaining chicken; transfer to plate. Let chicken cool slightly and discard skin.
Add remaining 2 tablespoons oil, mushrooms, onions, thyme, and ¼ teaspoon salt to pan. Cover and cook over medium heat until mushrooms are softened, 5 to 10 minutes. Uncover and continue to cook until mushrooms are dry and browned, 5 to 10 minutes.
Stir in garlic and cook until fragrant, about 30 seconds. Stir in flour and cook for 1 minute. Slowly whisk in wine, scraping up any browned bits and smoothing out any lumps; transfer to slow cooker.
Stir broth, carrots, soy sauce, and bay leaves into slow cooker. Nestle browned chicken with any accumulated juice into slow cooker. Cover and cook until chicken is tender, 4 to 6 hours on low.
Transfer chicken and vegetables to serving platter and tent loosely with aluminum foil. Let braising liquid settle for 5 minutes, then remove fat from surface using large spoon. Discard bay leaves. Stir in cream and tarragon and season with salt and pepper to taste. Spoon 1 cup sauce over chicken and serve with remaining sauce.