Using your hands or a rubber spatula, gently mix garlic, chicken, soy sauce, ginger, 4 scallions, and ½ cup broth in a medium bowl just to combine. Scooping out by the tablespoonful, form mixture into 1”-diameter meatballs.
Heat oil in a large skillet over medium-high heat. Add meatballs and cook, turning occasionally, until golden brown all over, 8–10 minutes. Transfer to a plate.
Combine broccoli and red pepper flakes in same skillet, season with salt and pepper, and cook over medium-high heat until broccoli is bright green and crisp-tender, about 5 minutes. Add meatballs and remaining 1½ cups broth. Bring broth to a boil, reduce heat, and simmer until meatballs are cooked through and broccoli is tender, 5–8 minutes. Serve sprinkled with more scallions.