Preheat oven to 350°F.
Melt 1 Tablespoon butter in a 9×13-inch baking pan. Add chicken breasts to pan and bake for 30 minutes.
In a medium saucepan, melt remaining 3 tablespoons butter. Add flour and stir until well blended. Add chicken broth and wine. Cook, stirring constantly over medium high heat, until sauce thickens and bubbles. Add and stir into the sauce: cheese, drained artichoke hearts, mushrooms and green onions.
Remove chicken from oven. Pour off excess liquid. Cover chicken breasts with sauce and bake 30 minutes more. Serve over rice.
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