Fresh herbs and sweet carrots update this old-fashioned skillet dinner. To retain its fragrance, add the thyme just before serving. Skillet Cornbread, with thyme instead of sage, is a natural accompaniment.
Season the chicken lightly with salt and pepper and then generously with paprika. In a heavy large frying pan over medium-high heat, warm the oil. Add the chicken and cook until brown, about 2 minutes on each side. Transfer the chicken to a plate. Add the onion to the frying pan and stir. Add the potato quarters and carrots. Sprinkle with salt and pepper and sauté until the vegetables are beginning to brown, about 5 minutes. Add the flour mix and stir to coat. Gradually mix in the broth and vermouth. Bring to a boil, stirring frequently. Return the chicken to the pan and bring to a boil.
Cover the pan, reduce the heat to medium-low, and simmer until the chicken and vegetables are cooked through, stirring and turning the chicken over occasionally, about 25 minutes. Mix in the thyme. Taste and adjust the seasoning. Divide the chicken and vegetables among 4 warmed plates and serve right away.