Heat vegetable oil in a large skillet or pot over medium-high heat.
Add the curry paste and cook for 1 minute, stirring constantly, until fragrant.
Add the chicken pieces and cook for 5-6 minutes, stirring occasionally, until mostly cooked through.
Pour in the coconut milk, fish sauce, and brown sugar, stirring to combine.
Add the sliced bell pepper and simmer for 12-15 minutes until the sauce thickens slightly and chicken is fully cooked.
Remove from heat and stir in fresh basil leaves.
Serve over jasmine rice and enjoy immediately.