To make soup, Warm oil in large pot over medium heat. Add carrots, onion, garlic, 1/2 tsp salt, and red pepper flakes and saute, stirring, until vegetables begin to brown, (about 15 minutes. Stir in broth, scraping up any bits stuck to bottom of pot. Cover and simmer until carrots are tender, stirring occasionally (about 30 minutes).
Make meatballs: Mix all ingredients in a medium bowl, season with salt an pepper, and roll into 1 inch balls, spreading them out on a platter.
Puree soup in a blender, or with an immersion blender, until smooth. Return soup to pot. Bring to a simmer. Add meatballs an let simmer until meatballs are cooked through (about 10 minutes). Add spinach and simmer until wilted (about 1 minute). Season with salt and pepper to taste.