Make the carbonara sauce. In a small mixing bowl or glass measuring cup, whisk eggs, egg yolks, and 1 cup of the Parmesan; season with a large pinch of salt and plenty of freshly ground black pepper. Set aside.
Cook pasta. Bring a large pot of salted water to a boil. Add pasta to boiling water and cook until al dente, according to package directions. Just before the pasta has finished cooking, use a soup ladle to scoop out 1 cup pasta water.
Steam and sauté mushrooms. While the pasta is cooking, heat the olive oil in a large skillet over medium heat. Add the mushrooms and season with 1/2 teaspoon salt. Cover, and cook over medium heat until mushrooms soften and release their liquid, about 5 minutes. Remove the lid and raise the heat to high. Add the shallot and cook, stirring frequently, until the liquid evaporates and the mushrooms are golden-brown, about 5 to 8 minutes more.
Season and deglaze. Stir in the thyme and smoked paprika. Pour in the wine and cook until the pan is almost dry, scraping up the bits that stick to the pan with a wooden spoon.
Toss pasta with mushrooms and sauce. Remove the pasta from the heat and drain. Return the pasta to the pan and toss with the mushroom mixture and parsley until well mixed. Pour the carbonara sauce over the top and continue tossing with two wooden spoons or tongs until well mixed. Thin with some of the reserved pasta water to the desired consistency.
Finish and serve. Season to taste with freshly ground black pepper and salt. Serve pasta, topped with the remaining cheese.
Food Safety Note: Raw eggs may pose a danger to pregnant women, babies, young children, the elderly, or anyone whose health is compromised. If you are at all concerned about food safety we recommend using pasteurized eggs for this recipe.
Parmesan Cheese: You can find vegetarian parmesan cheese as specialty grocers like Whole Foods.