Preheat the oven to 400°F (200°C).
In a large mixing bowl, whisk together the balsamic vinegar, honey, olive oil, minced garlic, salt, and pepper. Add chicken thighs and toss to coat well. Let marinate while you prepare the other ingredients.
Peel and chop the potatoes into even chunks. Place them in a large saucepan and cover with cold water. Bring to a boil and cook for 15-20 minutes, or until fork-tender. Drain and return them to the pot.
Add milk, goat cheese, butter, salt, and pepper to the potatoes. Mash until smooth and creamy. Set aside.
On a baking sheet, toss the carrot sticks with olive oil, salt, pepper, and thyme (if using). Spread them out in a single layer.
Remove the chicken from the marinade and place it on a separate baking sheet. Pour any remaining marinade over the chicken.
Place the baking sheet with the chicken and the sheet with the carrots in the preheated oven. Roast the chicken for 30-35 minutes or until the internal temperature reaches 165°F (75°C) and the carrots are tender (about 25-30 minutes).
When everything is cooked, serve the chicken thighs with a side of goat cheese mashed potatoes and roasted carrots. Garnish with fresh thyme if desired.