In a food processor, combine 2 tablespoons olive oil, garlic 1/2 of the basil, almonds and tomatoes. Pulse and chop until almost blended. The resulting pesto will be a little chunky and not too wet.
Place the pesto in a bowl and stir, adding another tablespoon olive oil and the grated pecorino. Season, to taste, with salt and pepper. Set the sauce aside so that the flavors will meld.
In the meantime, bring 6 quarts water to a rolling boil and add 2 tablespoons salt. Add the spaghetti and cook according to the package direction, until cooked, but still al dente. Drain well. Add the pasta to the serving bowl and toss to combine.