Place the chicken and chopped garlic in the slow cooker insert in a single layer.
In a bowl, whisk together tamari (or soy sauce), ketchup, honey, chili paste, rice vinegar, and black pepper until smooth.
Pour the sauce over the chicken, then add the broccoli florets around and on top of the chicken.
Cover and cook on low for 3–4 hours (or on high for 1–2 hours) until the chicken is cooked through and tender.
Remove the lid, set the slow cooker to high, and cook uncovered for about 25 minutes to reduce and thicken the sauce.
Serve the chicken and broccoli over bowls of rice, then top with chopped scallions/cilantro and toasted sesame seeds.