Season chicken with ½ teaspoon salt and ½ teaspoon pepper. Heat 2 tablespoons oil over medium-high heat in a large Dutch oven/soup pot. Once hot, add the chicken and sear until golden, about 2 minutes per side. Remove chicken to a plate but leave the drippings. Add onions and carrots to the drippings and sauté over medium-high heat for 4 minutes. Add curry paste, ginger, garlic and sauté for 2 minutes or until the onions are softened. Add chicken back to the pot followed by chicken broth, coconut milk, diced tomatoes, soy sauce, fish sauce, brown sugar, bay leaves, ½ teaspoon salt and ½ teaspoon pepper. Cover and bring to a boil, then reduce heat to a gentle simmer (covered) for 15 minutes or until chicken is tender enough to easily shred with two forks.
Remove chicken and let rest until cool enough to shred. Meanwhile, add bell peppers, mushrooms and rice to the pot. Simmer approximately 8-12 minutes or until rice is cooked. *Note: Cooking time may vary depending on how long it takes to bring your soup to a boil so I suggest checking the rice occasionally until done.
Remove from heat and discard bay leaves. Stir in the shredded chicken, basil, cilantro, lime juice, and Asian chili sauce to taste. If you would like a less “chunky” soup, stir in additional broth or coconut milk. Garnish servings with desired toppings.