Put a large pot of water on high heat, covered, and bring to a rolling boil.
Meanwhile, dump all of the tomatoes into a large sealable bag, squeeze out all the air and close. Using the palms of your hands, squash the tomatoes. Remove them from the bag, leaving their juices and seeds behind and roughly chop the flesh.
Add the pasta to the boiling water and cook to al dente – usually about 9 – 10 minutes.
When the pasta has cooked about 5 minutes, put 1 tablespoon of oil, the garlic, tomato paste and chili in a large pan and heat over low heat. Stir it around until the tomato paste melts down and the garlic becomes fragrant—about 3 minutes.
Now the pasta should have just a minute or so left. Dump half of the tomatoes into the pan with the salt and toss about. Take out about ¼ cup of the pasta water and add to the pan and let it bubble up (but take more water just in case).
Strain the pasta and shake very well to free it of any water. Dump into the tomato sauce and toss to combine.
Once it has absorbed most of the liquid add the remaining half of the tomatoes and olive oil, lemon juice and olives. Shut off the heat and toss everything to combine.
Transfer to two wide bowls and scatter with torn basil leaves and a little Parmesan (and remove the whole garlic cloves if that’s the route you took).