Season the chicken with salt and pepper. In a large (10-inch) saute pan, heat the oil over medium heat. Stir in the garlic and onions, season with salt, and cook gently until softened but not browned, about 10 minutes; stir occasionally and encourage the onions to break apart a bit. With a slotted spoon, transfer the vegetables and any small bits of garlic or onion to a plate.

If the pan looks dry, add another 1 tablespoon oil. Increase the heat to medium high and when the oil is hot, add the chicken, skin side down. Cook until well browned on both sides, 5 to 7 minutes per side. Transfer to a plate. Pour off and discard the oil in the pan. Return the pan to the heat and pour in the broth or water. Scrape the pan with a wooden spoon to release any browned bits and cook until the liquid has reduced to about 1/4 cup. Lower the heat to medium and return the garlic, onions, and chicken to the pan. Pour the marinara over the chicken and tuck in the thyme and lemon slices. Cover and simmer for 15 minutes, turning the chicken once halfway through. Stir in the olives and simmer, covered, until the chicken is cooked through (the jucies should run clear when pricked). Another 10 to 15 minutes.

Empty a can of tomatoes with their juices into a food processor; pulse until coarsely chopped. Transfer to a bowl and repeat with the remaining two cans of tomatoes (or empty all the tomatoes into a large bowl and break them up with your hands).

Heat the oil in a large (at least 4-quart) saucepan or Dutch oven over medium-low heat. Add the garlic or onion. Cook, stirring often, until the garlic or onion is golden and softened, 5 to 10 minutes for garlic (don't let it burn); about 20 minutes for onion. Stir in the red chile flakes, let them heat for 15 seconds to release their flavor, and then pour in the tomatoes with their juices. Bring to a boil over medium high, stirring frequently, and then reduce the heat to maintain a simmer. Add the salt and basil and simmer, stirring occasionally, until the sauce is full-flavored and has reduced to a thick, saucy consistency, 20 to 40 minutes (some brands of tomatoes take longer to cook down). Pools of orange may appear on the surface, which means the sauce is done. Taste and add the pinch of sugar only if the sauce lacks the natural sweetness of perfectly ripe, fresh tomatoes.