On a clean work surface, mound the flour and make a well in the center. Crack the eggs into the well, add olive oil and salt. Using a fork, gently whisk the eggs, gradually incorporating flour from the walls of the well until a shaggy dough forms. Knead by hand for 8–10 minutes until smooth and elastic. Wrap in plastic and rest at room temperature for 30 minutes.

While the dough rests, melt 2 tablespoons of butter in a large skillet over medium-high heat. Add the mushrooms in a single layer (work in batches if needed) and sauté without stirring for 2–3 minutes to get a golden crust. Stir and cook until tender and any liquid evaporates, about 5–7 minutes. Reduce heat to medium, add garlic and cook 1 minute until fragrant. Pour in the wine and let it reduce by half, about 2 minutes.

Pour in the heavy cream and simmer gently for 3–4 minutes until slightly thickened. Stir in the grated Parmesan, remaining 1 tablespoon butter, and season with salt and pepper. Keep warm on low heat.

Divide the rested dough into 4 equal pieces. Working with one piece at a time (keep the rest covered), flatten into a rectangle. Using a pasta machine or rolling pin, roll into thin sheets (setting 6 on a machine, or about 1/16 inch thick). Dust with flour and cut the sheets into 3/4-inch-wide ribbons. Toss the pappardelle with a little semolina or flour to prevent sticking.

Bring a large pot of generously salted water to a boil. Cook the fresh pappardelle for 2–3 minutes until al dente and the eggs in the dough are fully cooked (safe internal temperature 160°F/71°C). Reserve 1 cup of pasta water, then drain the pasta.

Add the drained pappardelle to the skillet with the mushroom ragù. Toss gently over low heat, adding a splash of reserved pasta water if needed to loosen the sauce until it coats each ribbon. Divide among warmed bowls, top with extra Parmesan and parsley, and serve immediately.