Whisk flour, chestnut flour, and salt in a bowl; stir in 3 tablespoons olive oil, eggs, and egg yolks to make a smooth dough. Form dough into a ball, wrap in a cloth towel, and let rest for 1 hour.
Roll pasta into thin 5x12-inch sheets on a floured work surface; cut sheets into 1x12-inch ribbons. Let pappardelle pasta dry on a kitchen towel sprinkled with flour while you complete remaining steps.
Melt butter in a large skillet over medium heat; cook and stir leeks, shallot, and garlic until white part of leeks becomes translucent, about 5 minutes. Heat 1 tablespoon olive oil in a separate skillet over medium heat; cook and stir mushrooms in hot olive oil until the liquid evaporates and mushrooms turn golden brown, 10 to 15 minutes. Mix mushrooms into leek mixture.
Stir white wine into mushroom-leek mixture and scrape up and dissolve any browned bits of food on the bottom of the skillet. Bring to a boil, reduce heat to low, and simmer mushroom-leek mixture until mushrooms are tender, about 5 minutes.
Bring a large pot of lightly salted water to a boil; cook the pappardelle pasta at a boil until tender and pasta floats to the top of the water, about 5 minutes; drain. Transfer pasta to a serving bowl, toss lightly with mushroom-leek mixture and parsley, and top with Parmigiano-Reggiano cheese.
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