Dry vermouth is the surprise star in this recipe. While it performs similarly to a dry white wine, the herbs and botanicals used to make vermouth add a more complex flavor that complements the roasted vegetables on the plate. Since Vermouth is a wine, store it in the refrigerator after opening to keep it fresh.
Preheat the oven to 450˚ F. Toss the carrots and fennel with 2 tablespoons olive oil on a rimmed baking sheet. Spread in a single layer and season with salt and pepper. Roast, stirring halfway through, until tender and lightly browned, about 20 minutes. Add the asparagus, season with salt and roast until bright green and crisp-tender, 5 to 8 minutes.
Meanwhile, heat the remaining 1 tablespoon olive oil in a large ovenproof skillet over medium heat. Season the chicken on both sides with salt and pepper, add to the skillet and cook until well browned on the bottom, about 5 minutes. Flip and cook until browned on the other side, 3 to 5 more minutes. Carefully add the vermouth to the skillet and transfer the skillet to the oven. Roast until the chicken is just cooked through, 15 to 20 minutes.
Remove the chicken to a cutting board to rest, about 5 minutes. Return the skillet to medium-high heat. Add the heavy cream and mustard to the pan juices, scraping up any bits from the bottom of the pan. Simmer until slightly thickened, about 2 minutes. Season with salt and pepper and stir in the parsley and vinegar.
Divide the vegetables among plates. Slice the chicken and add to the plates; spoon the pan sauce on top.