Whisk the sauce: In a medium bowl, whisk together the honey, soy sauce, minced garlic, ginger and rice vinegar until the honey dissolves and the mixture looks smooth.
Layer the vegetables: Spread the halved potatoes and sliced carrots evenly across the bottom of the slow cooker so they sit in the sauce and cook through evenly.
Add the chicken: Place the chicken thighs on top of the vegetables in a single layer, then pour the honey garlic sauce evenly over everything.
Cover and cook: Cover the slow cooker and cook on low for 6 hours or on high for 3 to 4 hours, until the chicken is fully cooked and shreds easily with a fork.
Add the broccoli: During the last 30 minutes of cooking, stir in the broccoli florets so they soften slightly but keep their bright green color.
Remove the chicken: Transfer the chicken to a cutting board and shred it into large pieces with two forks, or leave the thighs whole if you prefer.
Make the slurry: In a small bowl, stir the cornstarch into the water until no lumps remain, then pour it into the slow cooker liquid.
Thicken the sauce: Stir the sauce and let it sit uncovered on high for 10 to 15 minutes until it thickens into a glaze that coats the back of a spoon.
Return the chicken: Add the shredded chicken back into the slow cooker and toss gently so every piece gets coated in the thickened sauce.
Garnish and plate: Spoon the chicken and vegetables over steamed rice, then sprinkle with sesame seeds and sliced green onions before bringing it to the table.
