Bring the water to a boil in a medium saucepan over medium high heat. Slowly add the polenta, stir with a wooden spoon, and add 1 tsp salt. Reduce the heat to medium and cook, stirring occasionally, until the polenta is tender and full cooked, about 20 minutes. Add the milk, 2 tbsp of the butter, and the Parmigiano-Reggiano cheese to the polenta and stir together over medium-low heat until just warmed through and soft enough to drop easily from a spoon, a few minutes more. Cover to keep warm.

Meanwhile, heat a large skillet over medium heat. Add the olive oil and mushrooms and cook until they begin to brown and release their liquid, about 5 minutes. Raise the heat to medium-high and cook, stirring occasionally, until the mushrooms are crispy but still juicy, seasoning with salt and pepper toward the end of cooking. Add the shallot, garlic and thyme and cook 4 minutes more. Add the half and half and the Cognac. Reduce the heat to medium and cook, stirring frequently, until the liquid thickens just slightly and coats the mushrooms evenly, a few minutes more. Discard the thyme sprig and stir in the remaining 2 tablespoons of butter. Taste and season with more salt and pepper as needed.

Spoon the polenta into bowls and top with the mushrooms and sprinkle with herbs if desired.