Wash the dried mushrooms. Soak the mushrooms in 1 ½ cups of lukewarm water for 30 minutes. Carefully remove the mushrooms, dip them in another bowl of clean water then, briefly rinse and lay on paper towels or a clean kitchen towel to remove excess moisture. Strain ½ cup of the mushroom water through a fine sieve with cheesecloth – ensuring all dirt and sand is strained out and reserve that water for cooking. The rest of the mushroom water should be discarded.

Heat a medium skillet on medium heat and add 1 tablespoon of olive oil. When hot, add the mushrooms and rosemary and a pinch of salt and pepper. Cook until the mushrooms are lightly browned, then remove from the pan and roughly chop into the size pieces desired.

In a medium pot, bring the remaining 3 cups of water, the ½ cup of reserved mushroom water, cream, and kosher salt just up to a boil, then add polenta slowly while whisking. Whisk the polenta every couple of minutes for the first 6-8 minutes. When not whisking, put the cover on the pot but not completely, allowing a small amount of steam to escape.

Cook for 30 minutes, stirring occasionally with a rubber spatula or wooden spoon, making sure the polenta does not stick to the bottom or sides. If the mixture gets too dry, add some more cream, milk or water in ¼ cup increments. When finished the polenta should be creamy and tender but have some mouthfeel.

Stir in the mushrooms, cheese, and butter. Taste and, if needed, re-season with salt and pepper.