Drain chickpeas, reserving liquid. Taste chickpea liquid to determine if you want to use it for the soup; it should have a mild flavor and no off taste. Pour into a quart-size measuring cup, and add enough water to equal 1 quart. If the bean liquid doesn't taste right to you, discard it, and measure 1 quart cool water. Set aside.
Heat 3 tablespoons oil in a large heavy-bottomed pot or medium Dutch oven over medium. When shimmering, add carrots. Cook; stirring occasionally, until slightly softened, about 4 minutes. Add onion and 1/4 teaspoon salt; cook, stirring often, until onions are golden in spots, about 10 minutes. Push onion mixture to one side of pan; add garlic, cumin, ginger, turmeric, cayenne, and remaining 1 tablespoon oil to empty side of pan. Cook, stirring spice mixture constantly, until fragrant and spices coat bottom of pan, 30 seconds to 1 minute.
Add chickpeas, reserved 1 quart of bean liquid and water, and remaining 1 teaspoon salt to pan. Partially cover, and bring to a simmer over medium. Reduce heat to low, and gently simmer, partially covered, until chickpeas and carrots are very tender but chickpeas still hold their shape, about 30 minutes. (You should have about 5 1/2 cups of the cooked bean mixture. If needed, add additional water to equal 5 1/2 cups.)
Transfer 1 cup chickpea mixture to a blender, taking care to get a good mix of solids and liquid. Process until smooth, about 40 seconds; return to pan. Return the soup to a gentle simmer over medium.
Gradually add spinach in batches, stirring until wilted. Stir in lime juice, and add additional salt and cayenne to taste. Ladle soup into bowls, and drizzle with olive oil.
