1Warm oil in a large pot over medium heat. Add carrots, onion, garlic, ½ tsp salt, and red pepper flakes and sauté, stirring, until vegetables begin to brown, about 15 minutes.
3Stir in broth, scraping up any bits stuck to bottom of pot. Cover pot and simmer until carrots are tender, stirring occasionally, about 30 minutes.
5Meanwhile, make meatballs: Mix all ingredients in a medium bowl, season with salt and pepper, and roll into 1-inch balls, spreading them out on a platter.
7Puree soup in a blender, or with an immersion blender, until smooth. If using blender, return soup to pot. Bring to a simmer.
9Add meatballs and let simmer until meatballs are cooked through, about 10 minutes. Add spinach and simmer until wilted, about 1 minute. Season servings with salt and pepper to taste.