Coat a 4- to 5-quart slow cooker with cooking spray; set aside. Coat a large nonstick skillet with cooking spray; heat skillet over medium-high heat. Cook chicken, half at a time, in hot skillet until brown. Transfer chicken to the prepared cooker. Pour Alfredo sauce over chicken; sprinkle with thyme.
Cover and cook on low-heat setting for 5 hours. Add broccoli. Cover and cook on low-heat setting for 20 minutes more.
Meanwhile, heat brown rice according to package directions. Serve chicken, broccoli, and sauce over rice.