Heat the oven to 500ºF. In a skillet large enough to hold all the vegetables, melt 1 Tbs. of the butter over medium heat. Add the mushrooms and sauté until they're lightly browned and any liquid they release has evaporated, 8 to 10 min. Transfer to a small bowl.
In the same skillet, melt the remaining 1-1/2 Tbs. butter with the olive oil over medium heat. Add the onion and sauté until slightly wilted, 3 to 5 min. Add the carrot and celery; continue sautéing until they start to soften, about 5 min. Add the sun-dried tomatoes, vinegar, minced rosemary, thyme, and reserved mushrooms. Stir briefly to combine, reduce the heat to low, cover, and cook, stirring occasionally, until the vegetables have absorbed all the liquid, 15 to 20 min. Season to taste with salt and pepper and let cool slightly.
Meanwhile, rinse the chicken well inside and out and pat dry thoroughly with paper towels. Combine 1 Tbs. salt with 1/2 Tbs. pepper and season the chicken inside and out with this mixture. Put the rosemary sprigs in the cavity. Set the chicken, breast side up, on a rack in a roasting pan. Using your fingers, carefully separate the skin from the chicken breast and use your fingers to stuff a thin layer of the vegetables under the breast skin (you'll use about 3/4 to 1 cup); be careful not to tear the skin. Stuff the remaining vegetables into the cavity. Tuck the wing tips back and tie the legs together loosely.
Roast the chicken on the lowest oven rack with the legs pointing to the back of the oven at 500ºF for 15 min. Reduce the temperature to 450ºF and roast until an instant-read thermometer in the thickest part of the thigh reads 170ºF, another 30 to 45 min.
Transfer the chicken to a cutting board, tent with foil, and let it rest for 10 min. Untie the legs and spoon the stuffing from the cavity into the center of a warm serving platter. Cut the chicken into quarters and arrange around the stuffing. Serve immediately.