Thinly slice the portobellos and shiitakes.

Heat a tablespoon of butter in a large skillet over medium heat. When the butter is melted, add the mushrooms and sautéed on medium heat until cooked, about 7-8 minutes. Add the balsamic vinegar about halfway through cooking to allow to reduce. Cover and set aside.

In a medium pot, bring milk and chicken broth to a boil. Add the polenta and reduce to a very low simmer. (You will want to stir the polenta frequently while cooking, because even at the lowest setting, when it begins to thicken, it can pop quite surprisingly and splatter.)

Cook the polenta over very low heat, stirring frequently, for 25 to 30 minutes.

While the polenta is cooking, grate the cheese. When the polenta is fully cooked, add the grated cheese and stir to combine until creamy.

Add a generous amount of creamy, cheesy polenta to each plate and top with the sautéed mushrooms.