Bring the water plus 2 teaspoons of salt to a boil. Slowly add in the polenta, stirring constantly. Reduce heat and cook, stirring occasionally, until thick, about 30 minutes. Stir in the goat cheese until melted through and creamy. Remove from heat and stir in 2 tablespoons olive oil. Adjust seasoning with salt and pepper if desired.

While the polenta is cooking, prepare the mushrooms: Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the sliced mushrooms and thyme. Cook until browned, the mushrooms have released their liquid, and the liquid has evaporated, about 8 minutes. Add 1 teaspoon salt and ½ teaspoon pepper. Add the sherry and cook, stirring, until evaporated. Serve the mushrooms on top of the polenta.