It's probably a good idea if you wear latex gloves for this job, as the artichokes do tend to make your hands go black and they can be very prickly.
Cut the stalks of the artichokes to about 2" from the base of the heart and then cut across the top leaves. One by one, pull the tough outside leaves off. When you have striped the leaves back to the tender ones, stop and remove the choke with a teaspoon, inserting it into the center of the flower and carefully turning it. This will remove all the hairy spines. (Once you've done one you'll know exactly how to insert the spoon in the cleanest way, not breaking the sides of the choke.) Now remove the stringy outer skin on the stalk with a peeler. Put the artichokes into water with lemon juice squeezed into it to stop them going brown.
I know this all sounds like a "bit of a bother", but I think it's worth it, and you do get quicker at it; each artichoke should only take about a minute with good practice!
Cut the prepared artichokes into quarters. Cook gently in a little olive oil in a thick-bottomed pan with a good pinch of salt for about 4-5 minutes with the lid on. Remove the lid and add half the garlic and half the thyme. Allow the garlic to soften. Squeeze in the lemon juice and continue to cook until the moisture has cooked away. Remove from the heat while you wash your tomatoes.
Put the tomatoes into a bowl with a little olive oil, half the torn basil leaves, a pinch of salt and pepper and the dried chili to taste. Mix well, then spread evenly in an earthenware dish.
Scatter the artichokes and remaining garlic in and among the tomato mixture. Sprinkle over the rest of the thyme and basil. Season with sea salt and black pepper and drizzle with good olive oil. Place in the oven for about 40 minutes at 120 degrees.