Heat the vegetable oil in a large pot or Dutch oven over medium heat. Add in the shallots and cook until translucent, about 3-5 minutes.
Add in the chicken breasts and cook until the surface turns white.
Add in the red curry paste and sauté, stirring constantly, until the curry paste is fragrant and has deepened in color.
Slowly pour in the coconut milk.
Stir in the kaffir lime leaves, brown sugar, and fish sauce. Cover and simmer for 15 minutes, or until the chicken is cooked through.
Uncover, add in the red bell pepper, and simmer for another 1-2 minutes until the bell pepper is tender.
Serve over rice and garnish with cilantro and a juicy squeeze of lime.