In a large skillet, heat the olive oil over medium-low heat. Add the mushrooms and onion and cook, stirring, until softened, about 7 minutes. Add the beef, season with salt and pepper and cook over medium-high heat, stirring, until browned, 4 to 5 minutes. Stir in the tomatoes, lower the heat and simmer.
In a medium saucepan, bring 3 cups water and 3/4 teaspoon salt to a boil. Whisk in the polenta until thickened, about 3 minutes. Whisk in the cream, parmesan and 1/4 teaspoon pepper until smooth; pour into a large casserole dish.
Stir the green beans into the beef mixture and cook over medium heat, stirring, until warmed through, about 2 minutes. Season with salt and pepper. Spoon the mixture over the polenta and top with more parmesan.