Season the chicken thighs with salt and pepper. Place them in a skillet over medium heat and brown on both sides for about 3 minutes per side. Transfer the chicken to the slow cooker.
Add the diced onion, carrots, potatoes, and garlic to the slow cooker. Sprinkle the chopped rosemary over the vegetables and pour in the chicken broth.
Cover and cook on low for 6 hours. The chicken should become tender enough to shred easily with two forks.
Stir in the green beans, lemon zest, and lemon juice. Continue cooking on low for 30 more minutes until the green beans are tender but still bright.
Turn off the heat and slowly stir in the heavy cream until the broth turns smooth and creamy. Taste and adjust salt or pepper if needed before serving.