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Heat 2 tablespoons of the olive oil in large saute pan over medium high heat. Add the shallots and stir to coat in the oil. Cook until softened and transparent, about 5 minutes. Reduce heat to medium and add the mushrooms and salt stir to coat. Continue cooking until the mushrooms soften and deepen in color, they’ve released their moisture which has evaporated from the pan, and the mixture has become very fragrant, about 30 to 40 minutes more, stirring every few minutes. Set aside.
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While the mushrooms are cooking, you can prepare the base of the sauce. Pulse together the garlic, egg yolks, and 2 more tablespoons of the olive oil in a blender or food processor. Remove the center lid from the blender to create a small hole in the cap. With the blender running at medium low speed, *slowly* drizzle in the olive oil in the thinnest drizzle possible. You can stop the blender to wipe down the walls of it with a spatula if needed. You want the garlic bits at the bottom.
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Add the white wine in a drizzle, then the lemon juice, and then the thyme leaves, parmesan, and cheddar cheese. Blend until smooth. Add salt to taste. Set the sauce aside.
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Prepare the Barilla Classic Fettuccine according to the package directions. When the pasta is nearly done but not quite al dente, use tongs or a pasta claw to transfer the fettuccine to a large pot and immediately toss with the olive oil sauce. Add some of the hot pasta water, a tablespoon at a time, until the sauce becomes creamy, about 4 to 6 tablespoons. Add the roast turkey and the sauteed mushroom mixture and toss to combine. Sprinkle with the bread crumbs and serve immediately.