Heat 3 tablespoons of olive oil in a large pot over medium-high heat. Stir in the onion and garlic, and cook until the onion has turned golden brown, about 5 minutes. Stir in the tomato paste, cumin, kosher salt, black pepper, and 1/8 teaspoon of chili powder. Cook and stir 2 minutes more until the spices are fragrant.
Stir in the chicken broth, lentils, and carrot. Bring to a boil over high heat, then reduce the heat to medium-low, cover, and simmer until the lentils are soft, about 30 minutes.
Pour half of the soup into a blender, filling the pitcher no more than halfway full. Hold down the lid of the blender with a folded kitchen towel, and carefully start the blender, using a few quick pulses to get the soup moving before leaving it on to puree. Puree in batches until smooth and pour into a clean pot. Alternately, you can use a stick blender and puree the soup right in the cooking pot. Do not puree all of the soup, leave it a little chunky.
Stir in the lemon juice and cilantro, then season to taste with salt. Drizzle with olive oil and a sprinkle of chili powder to serve.