Bring a large pot of water to a boil. Add the lemon juice and 2 tablespoons of salt.
Snap off the stems of the artichokes and reserve. Trim the artichokes for all but their innermost leaves. Cut about 2 inches off the top of each artichoke and discard. Add the artichokes to the boiling water. Peel the tough outer skin off the stems and add the stems to the water. Place a small bread plate or other weight on top of the artichokes to keep them submerged. Lower the heat to a slow boil and cook until the artichokes' bottoms are just tender when pierced with the tip of a small knife through the bottom, about 15 minutes. Drain and let cool.
Trim the rest of the leaves off the artichokes and scrape out the chokes (the fuzzy centers) with a spoon. Cut the bottoms into quarters and the stems into 1/4-inch rounds.
Heat the oven to 375 degrees F.
Heat 2 tablespoons of the butter in a saute pan over medium-high heat. Add the onion and lower the heat to medium. Cook, stirring, until the onion is tender, about 7 minutes. Add the artichokes, season with salt and pepper, and continue to cook for an additional 3 to 4 minutes.
Strip the leaves away from the stems of the Swiss chard. Cut the stems into 1/2-inch pieces and the leaves into 1-inch pieces, and keep the stems and leaves separate. Heat the remaining butter in another saute pan over medium-high heat. Add the chard stems and cook for about 5 minutes. Add the leaves and continue to cook until the stems and leaves are tender. Season with salt and pepper. Drain the chard and add it to the artichokes and onion and toss to combine.
Put the artichokes, onion and chard into a 1 1/2-quart gratin dish. Drizzle the cream and sprinkle the Parmesan over the vegetables. Season with salt and pepper. Bake until the cheese is melted and the cream is bubbling, about 15 minutes. Raise the heat to 425 degrees F, sprinkle on the bread crumbs and bake again until bread crumbs are browned and the inside of the gratin is hot, another 7 to 10 minutes. Serve immediately.