Heat 1 Tbs. of the oil in a large pot over medium-high heat. Add the carrots and onion and cook, stirring occasionally, until the onion is soft and just golden-brown, about 10 minutes. With a rubber spatula, scrape the vegetables into a medium bowl and set aside. If necessary, wipe the pot clean.
Heat the remaining 1 Tbs. oil in the pot over medium-high heat. Add the pasta and cook, stirring often, until dark golden-brown, 3 to 4 minutes. Add the broth and stir, scraping the bottom of the pot to release any stuck-on pasta. Add the carrots and onions, kale, beans, lime juice, 1/2 tsp. salt, and 1/4 tsp. pepper and bring to a boil. Reduce the heat to medium low and simmer until the kale, carrots, and pasta are tender, 8 to 10 minutes.
Remove the pot from the heat, stir in the cilantro and season to taste with lime juice, salt, and pepper before serving.