Don't stir these tomatoes too much while they simmer, or you won't end up with the tasty plump pieces. The preserves are fantastic right off the spoon or as a topping for open-face serrano-ham sandwiches.
Bring a large pot of water to a boil. Score the tomato bottoms with an X. Plunge the tomatoes into the boiling water and blanch for 20 seconds to loosen their skins. Using a slotted spoon, transfer the tomatoes to a cutting board and let cool slightly. Peel the tomatoes. Halve them crosswise and remove the seeds. Cut out the hard cores.
In a large, heavy pot, combine the tomatoes, sugar, ginger and cinnamon stick. Set a strainer over the pot and squeeze the lemons into it; discard any seeds. Add the lemon halves to the pot and bring to a boil, stirring gently until the sugar is dissolved. Simmer over low heat, stirring occasionally, until the tomatoes are translucent and very tender, about 1 hour and 15 minutes. Discard the lemons and cinnamon stick.
Increase the heat to high and boil until the liquid is slightly reduced, about 15 minutes longer, stirring very gently to avoid scorching. Ladle the hot preserves into jars, cover and let cool.