Bring a large pot of lightly salted water to a boil. Add asparagus and cook until almost tender, 2–3 minutes. Transfer to a bowl of ice water; drain and pat dry. Reserve pot.
Meanwhile, bring chicken stock to a simmer in a large skillet over medium heat.
Return pot of water to a gentle simmer and add dumplings (one by one so they don’t crush each other), stirring occasionally, until they’re almost double in size and cooked through and tender, about 4 minutes (dumplings will quickly float to surface).
Transfer dumplings to pan; add butter and green garlic. Cook, tossing often, until sauce has thickened and garlic has slightly softened, about 3 minutes. Add asparagus, chives, and lemon juice and toss to warm asparagus; season with salt and pepper. Serve dumplings with more cheese and drizzle with olive oil.