Heat the oven to 375°F. Pare the artichokes down to the tender centers (or bottoms if using larger artichokes). Cut them in half.
In a medium nonstick frying pan, heat 1 Tbs. of the olive oil over medium heat. Add the shallots and cook about 1 minute. Add the artichokes, lemon juice, water, and sprinkle with 1/2 tsp. salt. Cover and cook until the artichokes are tender in the center when pierced with the tip of a knife, 10 to 20 minutes, depending on their size (for frozen artichokes, thaw them and cook 5 minutes). Uncover and continue cooking, stirring occasionally, until all the liquid has evaporated. Let cool.
In a large pot of boiling salted water, cook the spinach for 2 minutes. Drain in a colander and refresh with cold water. With your hands, squeeze out as much water as possible. Transfer the spinach to a cutting board and chop it finely. Crack the eggs into a large bowl, pour in the half-and-half, and whisk to combine. Season with about 1 tsp. salt and a few turns of the pepper mill. Add the cheeses, chopped spinach, basil, prosciutto, and the artichoke mixture and stir well.
Choose a baking dish or a small roasting pan large enough to hold an 8-inch nonstick frying pan. Add hot water to the dish to cover about one-quarter of the frying pan’s depth. This will act as a water bath for cooking the torta.
Heat the remaining 1 Tbs. olive oil in the nonstick frying pan over medium-high heat until a drop of the egg mixture sputters when added to the pan. Add the egg mixture and cook for 4 to 5 minutes over medium-high heat. With a spatula, lift the torta away from the edges of the pan to gauge its progress; when you see that the torta has browned nicely all around, remove the pan from the heat and immediately put it in the water bath to stop the browning. Put the pan and water bath in the oven and bake until the torta is firm in the center, 40 to 45 minutes.
Remove the frying pan from the water bath and turn out the torta, bottom side up, on a cutting board. Let cool for 10 to 15 minutes and cut into 1-inch chunks. Serve the torta pieces, top side up, at room temperature.