Preheat the oven to 350°F. In a large skillet, brown the chicken in butter. Transfer the chicken to an ungreased 13x9-inch baking dish, leaving the drippings in the skillet. Arrange the artichokes and mushrooms on top of the chicken. Editor’s Tip: The chicken will still be pink inside, but don’t worry about cooking it through. It will finish cooking in the oven.
Saute the onion in the pan juices until crisp-tender. In a small bowl, combine the flour, rosemary, salt and pepper. Stir the flour mixture into the pan until blended. Add the chicken broth, and bring the mixture to a boil. Cook and stir until thickened and bubbly, about two minutes.
Spoon the sauce over the chicken and vegetables. Bake, uncovered, until a thermometer inserted in the chicken breast reads 165°, about 40 minutes. Sprinkle with parsley just before serving. If desired, serve with pasta. Editor's Tip: To make the flavors really pop, you can finish the dish off with a shaving of fresh Parmesan and a squeeze of lemon juice.