Heat a large skillet over medium heat with a thin layer of extra-virgin olive oil. Add the onions, and garlic to the hot oil and sweat them until very soft and very light caramel in color, 20 to 30 minutes. Turn off the heat and reserve.
Heat a large Dutch oven over medium-high heat. Add a thin layer of oil to the pan. Pat the meat dry and season very liberally with salt and pepper. Brown the meat to a deep brown all over, 12 to 15 minutes.
Peel the potatoes and very thinly slice them lengthwise into planks. Slice the tomatoes working over a bowl to catch their juices.
Arrange 1/2 the onions over top of the meat, season with salt, pepper and 1 tablespoon thyme. Arrange 1/2 the potatoes over the onions and dress the potatoes with a liberal drizzle of extra-virgin olive oil, about 1 tablespoon, salt, pepper, rosemary, and a handful freshly grated cheese. Top the potatoes with 1/2 the tomatoes and their juices, and season with salt and pepper. Scatter a few leaves of torn basil. Repeat the layers. Do not add basil to the top layer of tomatoes.
Cover the pan and roast the meat in the oven until very tender, 4 hours. Cool and store for make-ahead meal.
To reheat: Reheat, covered, in a preheated 325 degree F oven until warmed through, 45 minutes to 1 hour. Take off the lid of the pot, drizzle the top of the meat with oil, sprinkle with cheese, and broil to lightly crust the top. Cut down through the layers of the potatoes, tomatoes, and onions to portion the meat, serve in shallow bowls with crusty bread and pan juices.