Preheat the oven to 375°F.
In a bowl, whisk together the wine, Dijon mustard, thyme, salt and pepper. Coat the chicken well.
Heat the olive oil in a large skillet or non-stick pan. Place the chicken skin-side down into the pan. Cook until the skin is golden brown and crispy.
Toss the potatoes and garlic in the remaining wine/Dijon mustard mixture along with the paprika, coat them well.
Once the chicken skin is golden brown, flip the chicken over and add the potatoes and any sauce into the pan. Spread the potatoes out a little bit so they aren't piled on top of each other, add the carrots to the pan and place in the oven.
Cook for about 20 minutes, then mix around the carrots and potatoes and add the mushrooms. Continue cooking until the chicken reaches an internal temperature of 160°F using a thermometer. Remove the chicken from the pan and set aside on a cutting board to rest.
Continue cooking the potatoes until they are nicely browned and the carrots and mushrooms are tender. Then remove from the pan and set aside.
Place the pan back on the stovetop over medium heat. Add in the broth and butter. Simmer until the sauce reduces and thickens to a smooth consistency. Taste and add salt if needed.
Remove the breast bones from the chicken and portion. Serve with potatoes, carrots, and mushrooms. Spoon the sauce over the chicken and garnish with thyme sprigs. Serve with buttered dinner rolls.