Bring 2 cups of the water to a boil in a medium saucepan over high heat. Add the polenta and salt, whisking constantly for 2 minutes. Decrease the heat and simmer, stirring from time to time, until the polenta is thick and pulls away from the side of the pan when stirred, 30 to 40 minutes.
About 15 minutes before the polenta is done, bring the remaining 4 cups water to a boil in a large saucepan. Add the vinegar and adjust the heat to a brisk simmer. Break the egg into a small bowl and pour it into the simmering water. Poach until the white turns opaque, 3 to 4 minutes. Remove the egg with a slotted spoon and place it on a clean kitchen towel to drain.
In a small skillet, heat 1 teaspoon of the olive oil over medium-high heat. Add the mushrooms and a pinch of salt and cook, stirring, until they’re tender and slightly browned, 4 to 5 minutes.
When the polenta is done, stir in the mushrooms, truffle oil (if desired), and remaining 1 teaspoon olive oil. Taste and adjust the seasoning, if desired. Transfer to a bowl, top with the poached egg and Parmesan, and serve.