In a large bowl, whisk together the mustard, vinegar, rosemary, and garlic. Once combined, slowly drizzle in the oil as you continue whisking. Add salt & pepper to taste. Reserve 2 tablespoons of the marinade, set the rest aside.
Place the chicken in the bowl with the marinade and toss to coat thoroughly. Cover and refrigerate for 30 minutes or overnight.
Preheat oven to 375°
Remove the chicken from the marinade, shaking off any excess. Discard remaining marinade.
Heat 1 tablespoon of olive oil in a cast iron pan over medium high heat.
Cook the chicken, 2 minutes per side, in the cast iron pan. Remove the chicken to a plate.
Place carrots, onion, and potato in the cast iron pan. Drizzle with reserved marinade. Toss to coat, then add salt & pepper to taste.
Bake for 30 minutes, or until vegetables are starting to become tender. Add the chicken back toi the pan. Spoon any liquid in the pan over the chicken. Bake for 15 minutes longer, or until chicken is cooked thoroughly and has an internal temp of 165°
Remove from oven, spoon any pan juices over the chicken and vegetables, and serve.