Preheat the oven to 375 degrees . In a large, heavy skillet, heat 1 tablespoon oil over medium heat. Season the pork with salt and pepper and cook in the skillet, turning, until browned, 10 minutes. Transfer to a 9-by-13-inch baking dish, reserving the skillet, and roast until the temperature registers 145 degrees on an instant-read thermometer, 12 to 15 minutes.
Meanwhile, heat the skillet over medium heat and cook the shallots and mushrooms in the remaining 2 tablespoons oil until softened, about 5 minutes. Add the chicken broth and simmer until reduced, about 3 minutes. Cover and remove from the heat.
In a saucepan, bring 3 1/4 cups water and the cream to a boil and whisk in the polenta. Lower the heat to medium low and cook, whisking, until tender, about 3 minutes. Stir in the cheese; season with salt and pepper. Cover and keep warm.
Transfer the pork to a cutting board and let sit for about 8 minutes, then slice on an angle and arrange on 4 plates. Top with the pan sauce and parsley and serve with the polenta.