For the chicken: Position a rack in the center of the oven and preheat the oven to 400 degrees F.
In a small bowl, mash together the butter, chopped rosemary, mustard, herbes de Provence, 1 1/2 teaspoon salt and 1/2 teaspoon pepper with a fork until smooth. Place the chicken, breast-side up, on a rack in a large roasting pan. Using your fingers, spread half of the butter mixture under the chicken skin. Spread the remaining butter mixture over the exterior of the chicken and in the cavity. Sprinkle the cavity liberally with salt and pepper and place the rosemary sprig inside. Tie the chicken legs together with kitchen twine. Roast until a meat thermometer inserted into the thickest part of the thigh registers 160 degrees F, about 1 hour 15 minutes. Let the chicken rest for 15 minutes.
For the bolognese: Place the dried porcini mushrooms and the hot chicken broth in a small bowl and let soak until the mushrooms are softened, about 20 minutes. Strain the mushrooms and reserve the broth.
In a food processor, combine the carrots, onions, bell peppers and garlic. Pulse until the vegetables are finely chopped. In a Dutch oven or large saucepan, heat the olive oil over medium-high heat. Add the chopped vegetables, oregano, thyme, 2 teaspoons salt and 1 teaspoon pepper. Cook until slightly softened, about 6 minutes. Add the porcini and fresh mushrooms, and cook until softened, about 5 minutes. Add the wine and cook, stirring frequently, until most of the liquid has evaporated, about 5 minutes. Add the reserved porcini broth and tomato paste and stir until dissolved. Bring the mixture to a boil, then reduce the heat to low and simmer until the liquid has reduced by half and the vegetables are tender, about 25 minutes. Remove the pan from the heat and stir in the mascarpone cheese until smooth. Season with salt and pepper.
Cut the chicken into thin slices and arrange on a platter. Spoon the sauce over the chicken and serve.
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