I make carbonara for my husband, John, after every big show he plays in NYC. This is an earthy and meat-free version of his favorite dish.
Bring a large pot of water to a boil for the pasta.
In a large skillet, heat the oil, about four turns of the pan, over medium-high. Add the mushrooms. Cook, stirring occasionally, until browned, 8 to 10 minutes. Season with the smoked paprika, salt, and pepper. Add the garlic. Reduce heat to medium. Cook, stirring often, until fragrant, 2 to 3 minutes. Add the wine. Cook, stirring often, until the liquid is absorbed, about 2 minutes. Reduce heat to low.
Salt the boiling water and add the pasta. Cook until 1 to 2 minutes shy of the package directions. Reserve about 1 cup of the pasta cooking water before draining.
In a medium bowl, beat the egg yolks and 1/2 cup of each cheese. Gradually whisk in the hot pasta water. Mix the pasta and egg mixture into the mushrooms. Remove from heat. Toss until the pasta is coated, 1 to 2 minutes. Mix in the herbs; season. Serve with more cheese.