
In a Dutch oven or similar large pot, heat the oil over medium heat. Add the bell pepper and carrots and cook, stirring occasionally, until starting to soften, about 5 minutes.
Add the ginger, garlic, white parts of the green onions, and curry paste and cook, stirring constantly, for 1 to 2 minutes, until fragrant and the paste starts to darken.
Add the chicken broth and increase the heat to high to bring to a simmer. Add the chicken breasts, then adjust the heat so the broth simmers gently. Partially cover the pot, and cook until the chicken is cooked through and registers 165°F on an instant-read thermometer, 12 to 15 minutes. Transfer it to a bowl and shred with two forks, or transfer to a cutting board and chop into bite-sized pieces.
Meanwhile, bring a large pot of salted water to a boil. Add the noodles and cook according to package directions. Drain.
Return the chicken to the pot. Stir in the coconut milk, soy sauce, and lime juice. Return to medium heat to warm through.
Divide the noodles between serving bowls and ladle soup on top. Garnish with reserved green onion tops and cilantro. Serve with lime wedges on the side.