Remove stems from mushrooms.

 

Cook carrot and garlic in butter several minutes. Add rice; cook 3 minutes, stirring occasionally. Add 1 cup wine; stir over low heat until wine is absorbed. Repeat with remaining wine.

 

Add chicken stock 1/2 cup at a time, stirring after each portion, until stock is absorbed. Stir in pepper. Add water 1/2 cup at a time, stirring constantly until rice is al dente. Stir in olives, scallion, and prosciutto; cook 1 minute.

 

Fill mushroom caps. Sprinkle with cheese.

 

Grill over medium coals 8 to 10 minutes or bake at 400 degrees F for 15 minutes.