For the polenta

Step 1

• Bring the water, milk, and salt to a low boil in a medium-size heavy saucepan set over medium-high heat. Sprinkle the polenta into the liquid, a little at a time, stirring with a whisk to prevent clumping. Do not add the polenta all at once. Continue stirring as the mixture thickens, 2 to 3 minutes.
  

Step 2

• Reduce heat to low and cook for 40 to 45 minutes, stirring every 4-5 minutes. If the polenta becomes too thick, add water or milk ¼ cup at a time. 
  

Step 3

• With a wooden spoon, stir in the butter and cheeses. Set aside to keep warm until ready to use.
  

For the mushrooms

Step 1

• Add the porcini mushrooms to a small pot with a fitted lid, and cover with about 1 cup boiling water. Cover the pot and allow the mushrooms to rehydrate for 20-30 minutes. 

  Remove the mushrooms and rinse well to remove any sediment. You may reserve or discard the mushroom stock left behind; if you decide to keep it, strain well with a fine-mesh sieve to remove any sediment. 

Step 2

• Melt the butter in a cast-iron skillet or other wide, heavy pan over medium heat. Add the shallots and garlic and sauté for up to a minute, allowing the ingredients to soften and become fragrant, but not to brown. Increase the heat to medium-high.
  

Step 3

• Add the porcini and fresh mushrooms and toss quickly to coat in the butter. Season generously with the salt and pepper, and add the thyme.

  Allow to cook, undisturbed, for 3 minutes, then toss and allow to cook for another 3-5 minutes, undisturbed, until the mushrooms have good color on them and have shrunk considerably, having released their liquid. 
  

Step 4

• Add the Marsala wine, and stir to combine. Turn the heat down to medium or medium-low, allowing the mushrooms to simmer gently for 5-7 minutes. Add the heavy cream and continue to simmer for 2-3 minutes. Taste, and correct seasoning. 
  

Step 5

• Serve the mushrooms on top of the polenta and garnish with parsley and cheese.