Trim off mushroom stems flush with caps. Rinse caps well; place gill side up in a lightly oiled 10- by 15-inch pan.
In a bowl, mix shrimp, green onions, mayonnaise, and cheese. Spoon shrimp mixture equally into mushroom caps.
Bake in a 400° oven until cheese is melted and mushrooms are flexible when pressed, about 12 minutes.
Meanwhile, in a 3- to 4-quart pan, combine broth, polenta, and tomatoes. Stir over high heat until boiling, then reduce heat to low. Stir often until polenta is very smooth to taste, about 3 minutes.
Spoon polenta equally into wide bowls. Set a stuffed mushroom on each portion. Add salt to taste.