Heat oil in a large skillet over medium-high heat. Add shallots; sauté 2 minutes. Add mushrooms; sauté 3 minutes or until tender. Stir in basil and next 5 ingredients (basil through tomatoes); cook 3 minutes or until thoroughly heated. Keep warm.
Combine vegetable broth and water in a large saucepan, and bring to a boil. Stir in the polenta. Reduce heat, and simmer until thick (about 5 minutes), stirring frequently. Stir in the cheese. Serve the tomato mixture over polenta.
I've made this dish many times and it's always wonderful. I like to use cremni or regular button mushrooms and add a good splash of red wine to the sauce. The type of onion used probably doesn't matter, and rosemary is a nice addition also. Served tonight with collard greens and a salad of romaine and artichokes. Coarse grated parmesan on top is great.
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