Heat a pot over medium heat and add the red curry paste, lemongrass paste, and coconut milk together. Stir until combined and let simmer for about 5 minutes.
Add in the chicken broth, carrots, mushrooms, and bell pepper then turn up the heat to bring it to a boil. When boiling, reduce the heat to low and simmer for 10 minutes.
Add in the cooked chicken breasts, soy sauce, brown sugar, and lime juice then stir until everything is combined. Simmer for another 5-10 minutes.
In a separate bowl mix together the cornstarch and cold water until it forms a smooth paste. Add this paste to the soup and stir until it thickens as desired.
Serve hot with a side of cooked rice or noodles.