Add water, milk or stock to a medium sized sauce pot and place over high heat. Whisk in the polenta so that it does not clump up. Bring to a boil and stir continuously until the polenta thickens and begins to bubble.
Turn the heat down to a low simmer and cook for about 20 minutes, stirring frequently with a spatula and making sure to scrape the bottom of the pan so it does not burn. Once the polenta is thick and tender, whisk in 3 tablespoons of the butter and season with salt.
In a large sauté pan set over medium heat, melt the remaining butter. Add the mushrooms and cook, stirring occasionally, until browned. Season with salt and pepper, to taste.
Pour in the beef broth and add the leftover Mississippi beef. Continue to cook, stirring, until beef is warmed through and mushrooms are incorporated. Serve warm, spooned over the polenta.