How to Make It
In a medium saucepan, bring the water to a simmer with a pinch of salt. Slowly whisk in the polenta until smooth. Cook over low heat, whisking often, until the polenta is tender and very thick, about 40 minutes. Stir in the butter, olive oil and Parmesan and season with salt and pepper. Cover and keep warm.
In a large skillet, bring 3 quarts of water to a boil. Reduce the heat to maintain a simmer and add the vinegar. Break the eggs into ramekins and, one by one, add them to the simmering water. Poach the eggs until the whites are firm but the yolks are still runny, about 3 minutes.
Preheat the broiler. Whisk the polenta over moderately high heat for 2 minutes to break up any lumps and heat through. Spoon the polenta into 6 individual gratin dishes and set the dishes on a baking sheet. Using a slotted spoon, transfer the poached eggs to the polenta, patting the bottom of the spoon with paper towels to dry the eggs. Sprinkle the shredded cheese over the eggs and broil for 30 seconds, or until the cheese is bubbling. Season with freshly ground pepper and serve.