In a saucepot, combine stock, water, lemongrass, garlic, chile and sliced ginger. Bring to a boil, reduce to a simmer and cover. Steep aromatics in broth 5-6 minutes. Remove and discard solids.
Cut remaining piece of ginger into thin matchsticks and add to broth. Turn off heat. Beat eggs with cornstarch. Stir broth vigorously to form a vortex and drizzle eggs into soup. Add scallions and chard, and serve at once.