Place the chicken breasts in the bottom of the slow cooker.
In a mixing bowl, whisk together the soy sauce, honey, sesame oil, ginger, garlic, chicken broth, and cornstarch until smooth.
Pour the sauce mixture over the chicken. Cover and cook on low for 5 hours.
Add the broccoli florets to the slow cooker. Stir gently to coat them in the sauce. Cook for 1 more hour until the broccoli is tender but still bright.
Remove the chicken and shred or slice it. Return it to the slow cooker and stir to combine with the broccoli and sauce.
Spoon the chicken and broccoli mixture over the cooked brown rice. Top each bowl with sesame seeds and sliced green onions.