Cook onion and ginger in medium skillet over high heat until charred, turning occasionally with tongs, about 10 minutes. Transfer to work surface. Cut ginger into thin rounds. Cut 2 inches from bottom of each lemongrass stalk and quarter lengthwise (discard remainder of stalks).
Place chicken in large pot. Add 3 quarts water; bring to boil. Skim foam from surface. Add green onions, peppercorns, and 1 teaspoon salt to pot. Add charred onion, ginger, and lemongrass. Reduce heat to medium-low. Partially cover pot; simmer 1 hour. Cool uncovered broth slightly. Strain and return to same pot (reserve chicken for another use). Spoon fat from surface of broth and reserve for matzo balls, if desired. (Can be made 2 days ahead. Chill soup and reserved fat until cold, then cover and keep chilled.)
Rewarm broth. Add cilantro and season to taste with salt and pepper. Place 2 warm matzo balls in each of 8 bowls. Ladle broth over.
*Available at Asian markets and in the produce section of some supermarkets.