Preheat oven to 400 degrees Fahrenheit (200 degrees Celsius).
In a small bowl, whisk together honey, minced garlic, chopped rosemary, soy sauce, and olive oil until emulsified.
Pat chicken drumsticks dry with paper towels and season with salt and black pepper.
Arrange drumsticks in a single layer in a 9x13 inch baking dish or on a parchment-lined rimmed baking sheet.
Pour the honey garlic mixture over the drumsticks, turning each piece to ensure uniform coating.
Bake for 40 to 45 minutes, basting the chicken with the glaze from the bottom of the pan at the 20-minute mark.
Continue roasting until the internal temperature reaches 165 degrees Fahrenheit (74 degrees Celsius) and the skin is dark golden brown.
Remove from oven and let rest for 5 minutes to allow juices to redistribute before serving.