Preheat your oven to 400°F (200°C). Line a large baking sheet with parchment paper or aluminum foil for easy cleanup.
In a small bowl, whisk together the melted butter, honey, minced garlic, olive oil, thyme, salt, black pepper, and red pepper flakes (if using) until well combined.
Place the carrot sticks in a large bowl. Pour the honey garlic butter sauce over the carrots and toss thoroughly until every piece is evenly coated.
Spread the carrots in a single layer on the prepared baking sheet, ensuring they aren't crowded. Roast in the preheated oven for 25-30 minutes, flipping halfway through, until the carrots are tender and caramelized around the edges.
Remove the carrots from the oven. Transfer them to a serving bowl. Drizzle with any remaining pan juices, then garnish with fresh parsley and sesame seeds (if using). Serve immediately while hot.