This is solid winter fare — meaty mushrooms, earthy Parmigiano Reggiano, nutty rice. The egg adds a luxurious element that butter alone can’t achieve. Even if you don’t have an Italian grandmother, you’ll feel like you do when you’re done with this dish. Nonna will have you stirring constantly so the rice doesn’t stick to the pot and absorbs the hot broth as it evaporates. Be sure to open that nice bottle of Nebbiolo, pour yourself a fat glass and keep it handy before you get started. You won’t be able to leave the stove for a while.
Wash and trim the mushrooms, separating caps and stems. Slice the caps and set aside. Add the stems to the broth and water in a medium saucepan and bring to a boil over high heat. Lower the heat to maintain a gentle simmer.
Heat the olive oil in a Dutch oven over medium heat. Sauté the prosciutto and shallots until the prosciutto has rendered its fat and the shallots are translucent, 2 to 3 minutes. Add the sliced mushrooms, 1/4 teaspoon of the salt and pepper to taste. Stir until the mushrooms soften, give up their liquid, and the liquid evaporates, about 8 minutes. Add the rice and cook for 1 or 2 minutes, stirring well to coat with the oil, then pour in the wine. When the wine has evaporated, remove the stems from the broth with a slotted spoon and ladle some broth into the rice, about 1/2 cup at a time, stirring constantly until the liquid is absorbed before adding more. Continue in this manner, adjusting the heat as necessary to keep a lively simmer, until the rice is tender yet firm to the bite, about 25 minutes. (Keep tasting the rice for doneness. You may not use all the broth. The rice should be creamy but not runny.) Stir in the remaining 1/2 teaspoon salt and as much black pepper as you’d like. Turn off the heat and stir in the butter to melt. Mix the yolk with the cheese and scrape into the risotto. Stir to distribute the yolk and melt the cheese. Taste and season with additional salt and pepper, if desired. Serve, passing more cheese at the table.
Adjust the heat to keep the risotto at a lively simmer as you cook. You don’t want the broth to evaporate too quickly (hard rice), but neither do you want the rice to be cooking too slowly (mushy rice).