Preheat the oven to 325 degrees F (165 degrees C). Melt 2 tablespoons of butter in a large skillet over medium heat. Saute mushrooms in butter until tender. Remove from the skillet, and set aside.
Place the chicken into the skillet, and pour the broth over it. Cover, and simmer over medium heat for 20 minutes, or until the chicken is cooked through. Reserve 1/2 cup of broth from the pan, and discard or save remaining broth for other uses. Remove the chicken to a 9x13 inch baking dish, and top with artichokes.
Melt the remaining 1/2 cup of butter in a skillet over medium heat, and whisk in flour until smooth. Gradually stir in the 1/2 cup of reserved broth, and half and half cream. Cook, stirring constantly, until thickened, about 5 minutes. Turn off the heat, and stir in the Parmesan cheese, rosemary, salt and pepper. Pour sauce over the chicken in the baking dish. Place sauteed mushrooms on the top.
Bake uncovered for 30 minutes in the preheated oven. Let stand for a few minutes before serving to let the sauce thicken.