Combine 1 lemongrass stalk, bottom third only, tough outer layers removed, 2 scallions, thinly sliced, 2 garlic cloves, smashed, one 1″ piece ginger, peeled, left whole, 2 Tbsp. soy sauce, 2 Tbsp. fish sauce, and 2 tsp. honey in a medium pot. Pour in 1 quart low-sodium chicken broth and 1½ cups water and bring to a simmer. Cook, stirring occasionally, 20 minutes.
Meanwhile, pulse 1 lb. large shrimp, peeled, deveined, in a food processor until mostly smooth; transfer to a small bowl. Mix in 2 garlic cloves, finely chopped, one 1″ piece ginger, peeled, finely chopped, 1 Tbsp. finely chopped cilantro, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, 2 Tbsp. water, and remaining 1 scallion, thinly sliced, until thoroughly combined.
Using damp hands, roll shrimp mixture between your palms into golf-ball-size balls (about 1½”); transfer to a plate as you go.
Remove lemongrass and ginger from broth; discard. Gently drop shrimp balls into broth; add 3 heads of baby bok choy (about 8 oz.), leaves separated, torn. Cook over medium-low heat, stirring occasionally, until shrimp balls are opaque throughout and bok choy is tender, about 3 minutes. Stir in 2 Tbsp. fresh lime juice and remaining 1 Tbsp. soy sauce; remove from heat.
Ladle soup into bowls and serve with lime wedges.