Fill a large pot with water and the juice of 2 lemons. Throw the juiced lemons into the pot with the acidulated water and bring to a boil.
Wash the artichokes under cold water. Cut the top quarter off the artichoke and trim off the spiked tips of the leaves with a pair of kitchen scissors. Cut the stems off so only the body is left and it sits flat. Place a clean, kitchen towel over the artichokes in the pot to help keep them submerged but don't let the towel hang over the sides of the pot near the flame!
Cook until tender, about 25-30 minutes, until a knife easily enters the base of the leaves. Remove from the pan and let cool. With your hands, gently pry open the artichokes to reveal the center choke section -- be careful not to pull the whole choke apart. Reach into the center and pull out the palest of the leaves until you have exposed the furry looking choke. Using a melon baller, carefully scoop out the choke from the center of the artichoke and discard. You should now have an empty center to stuff with the salad.
In a mixing bowl, combine the cherry tomatoes, red onion, parsley, basil, EVOO, salt and freshly ground black pepper. Combine thoroughly then evenly distribute the salad mixture among the artichokes, filling the exposed artichoke cavity and letting some spill over on to the plate.