In a large skillet, heat 2 tablespoons of extra-virgin olive oil over medium heat. Add the thinly sliced shallots along with a pinch of kosher salt. Sauté for about 3-4 minutes until softened and translucent.

Add the sliced portobello mushrooms to the skillet. Cook undisturbed for about 5 minutes, then stir occasionally for another 3-4 minutes until tender.

While the mushrooms are cooking, bring a large pot of salted water to a rolling boil. Add the pappardelle pasta and cook according to package instructions until al dente, usually 4-5 minutes for fresh pasta.

Combine the cooked mushrooms, garlic, fresh rosemary, red chili flakes, tomato paste, balsamic vinegar, and butter in the skillet. Cook for 2-3 minutes until fragrant.

Toss the cooked pappardelle into the skillet with the mushroom sauce. Mix gently to coat the pasta with the sauce, adding reserved pasta water if needed.

Transfer the pappardelle onto serving plates and generously grate Parmigiano Reggiano over each serving. Serve warm.