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Meatballs: Combine chicken, scallions, egg, miso, salt, and several grinds of pepper in a medium bowl. Mix vigorously with a fork until combined (it will seem a bit wet at first, so keep at it) and just beginning to feel slightly sticky; set aside.
Broth and Assembly: Heat oil in a large saucepan over medium. Add shallots and cook, stirring often, until softened, about 5 minutes. Add garlic and cook, stirring, until softened and just beginning to turn golden around edges, about 3 minutes. Add tomato paste and cook, stirring, until slightly darkened in color and beginning to sizzle in fat, about 2 minutes. Add cherry tomatoes, miso, and 4 cups water. Bring to a simmer, stirring vigorously and mashing miso against side of pot to dissolve, and cook, stirring occasionally, until most of the tomatoes have burst open, (it’s okay to help them along with a spoon), about 5 minutes.
Add green beans to saucepan and cook until tender, 6–8 minutes. Using 2 spoons, drop gumball-size balls of reserved meat mixture into simmering broth. Once all of the meatballs have been added, cook, stirring occasionally, until cooked through, about 5 minutes. Season with salt.
Ladle meatballs, vegetables, and broth into bowls and top with basil if desired.