If using a mortar and pestle: crush the garlic in a mortar with pinch of salt to form a paste. Add almonds and crush into a rough paste. Add the basil leaves and mint and crush them into small bits. Add in the tomatoes, then the cheese, followed by the olive oil. The final version should appear as a wet rough paste Season to taste with salt.
If using a food processor: pulse garlic with almonds until roughly chopped. Add the basil and mint, then cheese, tomatoes, and olive oil in process to a rough paste. Season to taste with salt.
Transfer 2/3 of the sauce to a large serving bowl. Do not heat the sauce.
Bring a large pot of salted water to a boil. Cook pasta until al dente. Transfer the pasta to the serving bowl, reserving some of the pasta water. Toss well, adding more extra-virgin, olive oil, and reserves pasta, water, a little bit at a time, until the creamy sauce emerges that’s dry, but not soupy. Season with salt to taste. If the sauce dries out, simply add more pasta water to loosen it.
Serve with remaining sauce, and extra freshly grated cheese and olive oil, as desired.