Break the chicken down. Use the legs, thighs, and breasts for this recipe. (I save the rest of make broth.)
Place flour in a shallow dish. Sprinkle the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and dredge in the flour.
Whisk water with 2 tablespoons of the leftover flour in a small bowl; set aside.
Heat 1 tablespoon oil in a large skillet over medium-high heat. Reduce heat to medium and add the chicken.
Cook, turning once or twice, until lightly browned on all sides, 5 to 7 minutes total. Transfer to a plate.
Add the remaining 1 tablespoon oil to the pan; reduce heat to medium-low.
Add mushrooms, carrots, onion, and rosemary and cook, stirring occasionally, until the vegetables are softened and browned in spots, about 5 minutes.
Add broth, wine, tomato paste, and the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Stir until the tomato paste is dissolved. Bring to a simmer.
Return the chicken and any accumulated juice to the pan. Cover, reduce the heat to maintain a simmer, and cook, stirring once or twice, until an instant-read thermometer inserted into the thickest piece of the chicken registers 165° degrees F (75° degrees C), 15 to 20 minutes
Transfer the chicken to a serving plate.
Increase the heat under the sauce to medium-high. Stir the water-flour mixture, add it to the pan, and cook, stirring, until the sauce comes to a boil. Continue cooking and stirring until thickened, about 1 minute. Serve the chicken with the sauce, sprinkled with parsley.