Start by bringing a large pot of salted water to a boil for cooking the pasta.

In a large Dutch oven, heat 3 tablespoons of olive oil over medium heat. Add the sage leaves and cook for about 15-30 seconds until crisp. Remove and drain on a paper towel.

Increase heat to medium-high and add the chopped mushrooms. Sauté for 6-10 minutes until browned and crispy. Season with salt, black pepper, and optional urfa chili flakes. Remove and set aside.

Add the dried pasta to boiling water and cook until two minutes shy of al dente. Reserve ½ cup of pasta water, then drain.

Return the pasta to the Dutch oven with ¼ cup reserved pasta water, bringing it to a boil and simmering until the pasta reaches al dente, about 2-3 minutes.

In a small bowl, whisk together ¾ of the grated Parmigiano-Reggiano, the egg, and the egg yolks. Slowly add 2 tablespoons of hot pasta water, mixing continuously.

Remove the Dutch oven from heat and slowly incorporate the cheese and egg mixture into the pasta while whisking to prevent curdling. Mix until the sauce thickens to a glossy consistency.

Gently fold in the sautéed mushrooms.

Taste the sauce and adjust seasoning if necessary. Divide pasta among serving bowls and top with reserved cheese and crispy sage leaves. Serve immediately.