3/4 lb (about 2-1/2 cups) cherry tomatoes, very ripe and sweet

12 large fresh basil leaves

1/3 cup of whole almonds, lightly toasted

1 plump garlic clove, crushed and peeled

1/4 tsp peperoncino or to taste

1/2 tsp coarse sea salt or kosher salt, or to taste, plus more for the pasta

1/2 cup extra-virgin olive oil

1 lb spaghetti

1/2 cup freshly grated Parmigiano-Reggiano or Grana Padano

1/2 tsp salt