Preheat oven to 375°F.
With a sharp knife, cut 2-3 slits into each drumstick and place into a sealable plastic bag. In a small bowl, whisk together olive oil, brown sugar, garlic powder, cumin, chili powder and paprika; add to ziplock bag with chicken. Turn bag over several times to coat. Place in refrigerator and marinate for 30 minutes.
Meanwhile, prepare the veggies: Slice carrots into 1/2 to 3/4-inch slices. Chop remaining veggies into pieces of the same size to allow for even roasting. Place all veggies into a large bowl and toss with thyme, vinegar and remaining 2 tablespoons oil. Spread out onto a baking sheet.
Remove chicken from marinade and place in a 13x9-inch baking dish in a single layer. Add to oven along with baking sheet holding veggies. Bake 25 minutes, then remove the veggies. Continue to bake chicken for 15 minutes longer or until no longer pink.
Slice peaches and serve alongside chicken and veggies.