Put extra-virgin olive oil, onions, and garlic into a large deep skillet and cook over medium heat until onions are soft, about 10 minutes. Add wine, cook for 1 minute, then add tomato juice, stock, and bay leaves and simmer until sauce has reduced by two-thirds, about 2 hours.
Meanwhile, bring a large pot of salted water to a boil over high heat. Cut leaves off chard stalks, halve lengthwise, and set aside. Cut stalks crosswise into 3"–4" pieces and set aside. Blanch chard leaves for 1 minute, transfer with a slotted spoon to a colander to let drain, then set aside. Working in batches, first boil chard stalks, then string beans, then carrots, then asparagus, then artichokes in same pot of boiling water, in that order, until tender, 3–4 minutes for the chard, 8–10 minutes for the beans, 12–14 minutes for the carrots, 4–5 minutes for the asparagus, and 12–14 minutes for the artichokes, transferring them as done to a colander to let drain, then setting them aside separately. Slice carrots crosswise into 1"-thick pieces and set aside. Squeeze out excess water from each chard leaf, forming an individual bundle, and set aside.
Pour olive oil into a wide deep pot to a depth of 2" and heat over medium heat until temperature reaches 265° on a candy thermometer. Put flour into a shallow dish and eggs into another shallow dish. Dredge chard leaf bundles in flour, dip in egg, then deep-fry until golden, about 1 minute. Transfer with a slotted spoon to a plate and set aside. Repeat dredging, dipping, and deep-frying process with chard stalks, asparagus, artichokes, brains, and sweetbreads, frying them until golden, about 2 minutes for the stalks, 30 seconds for the asparagus, 3 minutes for the artichokes, and 3 minutes each for the brains and sweetbreads, transferring them as done with a slotted spoon to separate plates.
Season tomato sauce to taste with salt. Add carrots and string beans to sauce, then add chard bundles and stalks, asparagus, artichokes, sweetbreads, and brains in layers. Simmer until warmed through, about 10 minutes. Discard bay leaves. Transfer to a large warm platter and serve.