Heat 1 tablespoon of olive oil in a large stock pot (or Dutch oven) over medium-high heat. Season chicken breast with salt and pepper, to taste. Place the chicken breast into the pot to brown on both sides 2-4 minutes. Time will be dependent on how thickly cut the breasts are. Internal temperature should read 165 degrees F using a meat thermometer. Remove to plate and cover to keep warm.
Add 1 tablespoon of olive oil to the pot to warm. Once heated, add carrots, peppers, and onions (not green onions). Sauté 3-4 minutes, or until the onions are translucent. Add minced garlic and ginger. Cook 30-60 seconds.
Add red curry paste, stock, tomatoes, soy sauce, basil, cilantro, and mushrooms. Stir well. Season with salt and pepper, to your taste preference (see notes below on sodium).
Bring to a boil then reduce heat to medium-low and gently simmer for 8-10 minutes. Shred chicken using two forks.
Add coconut milk, lime juice, lime zest, chopped green onions, and zoodles. Add shredded chicken back to re-warm.
Allow to cook 2-3 minutes, or until the zoodles have reached desired doneness. Keep in mind, they will continue to cook in the broth as it cools.
Serve garnished with green onions, red pepper flakes, freshly chopped basil and/or cilantro, and a lime wedge.