In a large skillet, heat the EVOO, three turns of the pan, over medium-high. Add the mushrooms and cook, stirring often, until browned, about 10 minutes. Add the asparagus, leek, thyme and garlic; season with salt and pepper. Cook, stirring often, until the leek softens, about 10 minutes. Add the sherry and stir until the liquid evaporates, about 2 minutes.
While the mushrooms and vegetables cook, bring the stock and 1 1/2 cups water to a boil in a large saucepan. Reduce the heat to low, then gradually whisk in the polenta. Stir until the polenta thickens, 3 to 5 minutes. Stir in the cheese, butter and nutmeg; season. Cover and turn off the heat. (If the polenta gets too thick as it rests, gradually stir in more hot water.)
In a large saucepan, bring 2 to 3 inches water and a splash white vinegar to a simmer over medium-low. Crack 1 egg into a small bowl or ramekin. Whisk the water in circles, then add the egg. Repeat with the remaining eggs. Poach about 3 minutes, then turn the eggs with a slotted spoon. Cook until the whites are set but the yolks are still runny, about 30 seconds more. Transfer the eggs to a plate.
Divide the polenta among bowls. Make a well, then fill with the mushroom mixture. Top each with a poached egg.