In a 4-to 5-quart slow cooker combine mushrooms, carrots, and onions. Set aside.
Lightly coat a large nonstick skillet with cooking spray; heat over medium-high heat. Sprinkle both sides of chicken thighs with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. Add chicken thighs to skillet; meaty sides down. Cook about 6 minutes or until brown, turning once. Add chicken to cooker.
Add wine to skillet; bring to boiling. Reduce heat; simmer for 2 minutes, using a wooden spoon to scrape up brown bits from bottom and sides of skillet. Remove from heat. Stir in broth, tapioca, vinegar, herbs de Provence, garlic, and the remaining 1/4 teaspoon salt and the remaining 1/4 teaspoon pepper. Pour over chicken.
Cover and cook on low-heat setting for 5 to 5 1/2 hours or on high-heat setting for 2 1/2 to 2 3/4 hours.