Bring water and broth to a simmer in a saucepan (do not boil). Keep warm over low heat.
Heat 2 teaspoons oil in a medium saucepan over medium-high heat. Add mushrooms; saute 5 minutes or until tender. Remove mushrooms from pan.
Heat 2 teaspoons oil in pan over medium-high heat. Add onion; saute 2 minutes or until tender. Add rice; cook 1 minute, stirring constantly. Stir in wine; cook 2 minutes or until liquid is nearly absorbed, stirring constantly.
Add broth mixture, 1/2 cup at a time, stirring constantly until each portion of broth mixture is absorbed before adding the next (about 20 minutes). Stir in mushrooms, cheese, and remaining ingredients.