Bring 6 cups of water and 1 teaspoon salt to a boil. Lower the heat to medium and slowly add the cornmeal in a shower, whisking constantly until the mixture thickens, 3 to 5 minutes. Change to a wooden spoon and continue to simmer, stirring occasionally, until the spoon stands in the polenta, 15 to 25 minutes.
Add the Parmigiano and 2 tablespoons of the butter and mix well. Season with salt and pepper. Immediately spread the polenta in a buttered 9-by 9-inch pan. Smooth the top with a rubber spatula and refrigerate. Combine the parsley and garlic and chop together until very fine.
In a large skillet, melt the remaining 2 tablespoons butter with the extra-virgin olive oil over high heat. Add the wild mushrooms and cook, stirring occasionally, until golden and the mushrooms and liquid has evaporated, 7 to 10 minutes. Remove the mushrooms and set aside. Add the chicken stock, cream, garlic, and parsley to the pan and simmer to reduce by half, 6 to 8 minutes. Season with salt and pepper.
Score the polenta into 3 by 4 1/2- inch squares. Halve each square diagonally to make 2 triangles. Remove from the pan and toss the polenta triangles carefully in flour to dust them lightly. Heat 1/2-inch of olive oil in a tiny piece of polenta turns golden on contact, 375 degrees F. Add a few of the polenta triangles and cook, turning occasionally, until golden on both sides. Drain on paper towels.
To serve, place 2 hot polenta triangles on each plate. Warm the mushroom ragout and spoon over the polenta. Serve immediately.