Season the chicken all over with the salt and pepper.
Heat the olive oil in a large stockpot over high heat. When the oil shimmers, add the chicken, skin side down, and sear until golden and crisp, about 5 minutes.
Flip the chicken, add the curry paste and carrots, and cook until fragrant, 1 to 2 minutes.
Stir in the coconut milk, chicken broth, fish sauce, and honey. Cover and bring to a boil.
Reduce the heat to medium-low, cover, and cook until the chicken is cooked through, about 15 minutes. Using two forks, shred the chicken directly in the pot. Stir in the greens.
Meanwhile, bring a large pot of salted water to a boil over high heat. Cook the noodles according to the package directions, until al dente. Drain.
Divide the soup among four bowls and top each bowl with the noodles, cilantro, chiles, and limes.