Bring a large pot of water (4-6 quarts) to a boil and add 3 tablespoons salt. Cook the pasta according to package directions, about 8 minutes for perfect al dente texture (test and taste to be sure). Scoop out and reserve 1/2 cup of the cooking water, then drain the pasta.
Make the sauce
While the pasta is cooking, heat the oil in an 6-8-inch skillet over medium heat. Add the almonds and garlic. Cook, shaking the pan once or twice, until the almonds start to toast and the garlic is fragrant, 3-5 minutes. Stir in the chili, roasted red peppers, tomato and salt and cook 2 more minutes.
Transfer the almond mixture to a food processor or blender. Add the cheese and 2 tablespoons of the pasta water. Process until a smooth sauce forms. Add more water, drop by drop, if needed to get a thick, saucy consistency.
Put the pasta in a large serving bowl and pour the sauce over. Gently toss until the strands are coated in the sauce. Sprinkle with additional Pecorino cheese and the basil leaves and serve. Buon appetito!
To roast red peppers: Roast the peppers on a grill, stovetop gas burner or under the broiler until blackened all over. Cool, then peel and remove the stem and seeds.