Preheat the oven to 425°. Rub the butter all over the chicken and set it in a medium cast-iron skillet. In a small bowl, mix the paprika with 1/2 teaspoon each of salt and pepper and sprinkle all over the chicken. Scatter the carrot, celery, onion, garlic and reserved chicken parts around the chicken and cook over high heat until sizzling, about 2 minutes.
Transfer the skillet to the oven and roast the chicken for 50 minutes, or until the skin is golden brown and crisp and an instant-read thermometer inserted in the thickest part of the thigh registers 170°. Transfer the chicken to a cutting board and let rest for about 10 minutes.
Add the wine to the skillet and cook over high heat, stirring, until reduced to 1/2 cup, about 5 minutes. Strain the pan juices into a bowl and skim off any fat.
Carve the chicken. Spoon the Farro with Shiitake Mushrooms onto plates and arrange the carved chicken on top. Garnish with thyme sprigs and serve with the pan juices.