Rinse the cherry tomatoes and pat them dry. Rinse the basil leaves and pat dry.
Drop the tomatoes into the blender jar or food processor bowl followed by the garlic clove, the almonds, basil leaves, and 1/2 tsp salt. Blend for a minute or more to a fine purée; scrape down the bowl and blend again if any large bits or pieces have survived.
With the machine still running, pour in the olive oil in a steady stream, emulsifying the purée into a thick pesto. Taste and adjust seasoning. (If you’re going to dress the pasta within a couple of hours, leave the pesto at room temperature. Refrigerate for longer storage, up to 2 days, but let it return to room temperature before cooking the pasta.)
To cook the spaghetti, heat 6 quarts of water, with 1 tablespoon salt to the boil in the large pot. Scrape all the pesto into a big warm bowl.
Cook the spaghetti al dente, lift it from the cooking pot, drain briefly, and drop onto the pesto. Toss quickly to coat the spaghetti, sprinkle the cheese all over, and toss again. Serve immediately in warm bowls