Heat the oven to 450°F.
Line a large sheet pan with parchment or foil (optional, but will make it easier to clean). (Note: The key to nice golden brown roasted vegetables is not to crowd them on the pan, so if it looks like your vegetables will be crowded, use two pans.)
While the oven is heating, chop the Brussels sprouts and carrots.
Toss the vegetables with 2 Tbsp cooking oil and season generously with some salt and pepper.
Spread out on the sheet pan.
Add the chicken thighs to the pan, nestling them between the vegetables.
Drizzle the tops of the chicken with 1 Tbsp cooking oil and then season the tops of the chicken thighs with some salt and pepper.
Transfer the pan to the oven and roast for 15 minutes to get the chicken and vegetables started cooking.
Whisk together mayonnaise, honey, and mustard. Divide into two small bowls.
After they have been cooking for 15 minutes, remove the chicken and vegetables from the oven. Use a spatula to turn the vegetables so that the cook evenly.
Brush the first half of honey mustard over the tops of the chicken thighs. (Reserve the second half.)
Return the sheet pan to the oven and continue cooking until the chicken is cooked through and the vegetables are tender, 10 to 15 minutes more. (The timing will vary depending on the size of the chicken thighs; when done, the chicken should register 165°F degrees in the middle).
Serve chicken and vegetables with the remaining honey mustard for dipping.