In a large skillet over medium heat, add the olive oil and heat until shimmering. Add the mushrooms and cook, stirring occasionally, until the mushrooms have shrunk considerably, are turning very golden brown and crispy, about 10-15 minutes. Season the mushrooms generously with salt and remove from the pan with a slotted spoon. Set aside for later.
Bring a large pot of heavily salted water to a rolling boil. Add the pasta and cook for 2 minutes less than the package suggests
While the pasta is cooking, add the chili flakes and tomato paste to the skillet. Cook, stirring constantly, for about 2 minutes.
Pour the wine over the tomato paste and cook, stirring constantly, until the wine has reduced significantly and the strong alcohol scent has dissipated, about 2-3 minutes.
Add the tomatoes, increase the heat to medium-high and simmer until the pasta is done cookng.
As soon as the pasta is finished, transfer it to the skillet using tongs (or drain the pasta, making sure to reserve some of the cooking water beforehand). Add the pecorino and toss the pasta in the sauce until it is thickened and cohesive, another minute or two. Add some of the reserved pasta water to adjust the consistency of the sauce, as needed.
Add the crispy mushrooms to the skillet and toss to incorporate into the pasta. Serve immediately, topping with more pecorino if desired.