Dice the onion, carrots, and red bell pepper into even pieces, mince the garlic, grate the ginger, and rinse and drain the chickpeas. Chop the cilantro, cut the lime into wedges, and measure the spices into one small bowl.
Heat the olive oil in a large Dutch oven over medium heat. Add the onion, carrots, and bell pepper with the salt and black pepper and cook 8 to 10 minutes, stirring occasionally, until the onion is soft and translucent.
Add the garlic, ginger, curry powder, turmeric, and cayenne. Stir constantly for 1 minute until fragrant and toasty, adding a splash of broth if the pan looks dry.
Pour in the diced tomatoes and scrape up any browned bits, then add the broth and stir. Nestle the chicken breasts into the liquid, bring to a boil, then reduce heat to low, cover, and simmer 20 to 25 minutes until the chicken reaches 165°F.
Transfer the chicken to a cutting board and rest 3 to 5 minutes. Shred with two forks or meat claws, then return the chicken and any juices to the pot.
Add the rinsed chickpeas, bring back to a gentle simmer over medium-low heat, and cook uncovered for 5 minutes. Taste the broth and note whether it needs more salt.
Shake the coconut milk well and stir it in until the broth turns creamy and golden. Warm over low heat for 3 to 5 minutes without boiling, then stir in the lime juice and adjust salt or cayenne to taste.
Ladle into bowls, making sure each gets plenty of chicken, carrots, and chickpeas. Top with fresh cilantro and serve with a lime wedge on the side.