In a large saucepan of boiling salted water, cook the orzo until it is al dente, about 8 minutes. Drain and rinse the orzo under cold water until cool.
Meanwhile, in a medium soup pot, heat the olive oil until shimmering. Add the chicken meatballs and cook over moderately high heat until they are lightly browned, about 4 minutes. Using a slotted spoon, transfer the meatballs to a plate. Add the garlic to the pot and cook over moderate heat until lightly golden, about 1 minute. Add the chicken broth, bring to a simmer and season with salt and pepper. Add the meatballs to the broth and simmer until they are cooked through, about 3 minutes.
Add the baby spinach and cooled orzo to the simmering broth and cook, stirring, until the spinach is wilted and the soup is piping hot, about 1 minute. Ladle the meatball-and-orzo soup into shallow bowls and serve.