Preheat the oven to 425 degrees F and prepare a parchment lined baking sheet with two sheets of parchment, set one aside and one on the cookie sheet.
Cut the potatoes in 1 inch cubes. Halve the brussels sprouts.
Peel and cut each carrot into 4 pieces. Peel the onion and slice it in half, cutting in large half circles. (See above for sizes)
Pat the drumsticks dry with a paper towel. Place the drumsticks on a lined cookie sheet, spreading them apart so there is space in between them.
Take the chopped potatoes, brussels sprouts, carrots, onion, and garlic and place them between the drumsticks on the cookie sheet.
Pour the olive oil evenly across the chicken and potatoes, then sprinkle with spices.
Mix the oil and seasoning into the chicken and vegetables with a pair of tongs or your hands so that all pieces are evenly coated (This is where food grade gloves come in handy.)
Optional: Place a second piece of parchment paper on top of the sheet pan. This is optional, but allows the moisture to stay locked in, resulting in more tender chicken.
Place the sheet pan in the middle rack of your oven and cook at 425 degrees for 1 hour. Bake covered for 40 minutes then uncovered for the last 20 minutes. If you want to get some extra crispiness on the tops of your drumsticks and vegetables, turn your oven to the broil setting and broil for 2-4 minutes once finished cooking.