The key to this sauce is a creamy emulsion of the oil, cheese, and pasta water. Toss and stir—and stir and toss—adding liquid freely, until it comes together.
Pulse toast in a food processor to fine crumbs; set aside.
Cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente; drain, reserving 1 1/2 cups pasta cooking liquid.
Meanwhile, heat chiles and 3 tablespoons oil in a large skillet over medium heat until sizzling, about 2 minutes. Add tomatoes, increase heat to medium-high, and cook, turning tomatoes occasionally, until blistered in spots, about 3 minutes. Continue to cook, lightly crushing tomatoes, until beginning to break down, about 3 minutes.
Add pasta, 1 cup pasta cooking liquid, and 1/4 cup Pecorino to skillet and cook, tossing and adding more liquid as needed, until sauce coats pasta, about 2 minutes.
Serve pasta seasoned with salt and pepper and topped with remaining Pecorino and reserved breadcrumbs.
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