Add the rinsed brown rice to the slow cooker. Arrange the chicken breasts on top, then pour in 6 cups of the chicken stock. Add the bay leaf and thyme.
Cover and cook on low for 5 to 6 hours or high for 2 1/2 to 3 1/2 hours, until the rice is tender and the chicken is cooked through. To prevent the chicken from becoming dry, check it early. (To check for doneness, insert an instant-read thermometer. The chicken is done when it registers 165 degrees F at the thickest part.) If the chicken finishes cooking before the rice, remove it first so that it does not overcook, then re-cover the slow cooker and continue cooking until the rice is tender. Remove the bay leaf and thyme stems and discard.
Place the chicken on a plate or cutting board and let it cool slightly. With two forks or your fingers, shred the chicken, then return it to the slow cooker. Stir in the lemon juice. If you'd like the soup thinner, add additional stock until you reach your desired consistency. Taste and adjust the seasoning as desired. Serve warm, sprinkled with additional fresh thyme.
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