Soak mushrooms in a medium bowl of hot water, covered, until soft, about 1 hour. Meanwhile, bring a medium pot of water to a boil over high heat. Add sweetbreads and lemon slices, reduce heat to medium, and simmer for 10 minutes. Drain, discarding lemon slices. To make sweetbreads easier to slice, arrange them in a single layer on a plastic wrap-lined baking sheet. Cover sweetbreads loosely with plastic wrap, then set another baking sheet directly on top and weight it down with several heavy, unopened cans. Refrigerate for 30 minutes. Slice sweetbreads on the bias into 1⁄2"-thick pieces and set aside.
Remove mushrooms from soaking liquid, trim off stems, then chop caps and set aside. Discard soaking liquid and stems. Whisk together stock, oyster sauce, sherry, soy sauce, and sugar in a small bowl and set sauce aside.
Heat oil in a large skillet over medium-high heat. Add onions, garlic, and ginger and cook, stirring constantly, until onions are soft, 3-4 minutes. Add sweetbreads and mushrooms, reduce heat to medium and cook, gently shaking skillet over heat, until sweetbreads are heated through, about 1 minute. Add sauce and simmer, stirring and spooning sauce and mushrooms over sweetbreads, until thickened, 1-2 minutes. Transfer to a warm serving platter and garnish with scallions.