Preheat your oven to 400°F and line a large baking sheet with parchment paper.
Pat the chicken drumsticks completely dry with paper towels to ensure the skin achieves maximum crispiness.
In a mixing bowl, toss the drumsticks with olive oil, sea salt, and black pepper until every piece is evenly coated.
Place the chicken on the baking sheet and roast for 35 minutes, turning once halfway through the cooking time.
While the chicken is roasting, whisk the honey, tamari, and minced garlic in a small saucepan over low heat for 3 minutes until the sauce thickens slightly.
Remove the chicken from the oven, brush each drumstick generously with the honey-garlic glaze, and return to the oven for 5-7 minutes until caramelized and dark golden brown.
Steam the broccoli florets in a basket over boiling water for 4-5 minutes until they are tender-crisp and bright green.
Plate the drumsticks alongside the steamed broccoli and sprinkle with sesame seeds before serving.