Place the chicken breasts in the slow cooker.
Add the diced onion and minced garlic on top of the chicken.
Whisk together the Dijon mustard, honey and softened butter, then pour the mixture over the chicken. Season with salt and pepper and tuck the rosemary sprig into the sauce.
Cover and cook on high for 3–4 hours (about 240 minutes) or until the chicken reaches an internal temperature of 165°F and is easily shredded.
Remove and discard the rosemary sprig. Transfer the chicken to a plate or cutting board, shred with two forks, and return the shredded chicken to the slow cooker. Stir to coat with the sauce.
Stir in a few dried chives for color if desired, keep warm until ready to serve, and serve over rice.