Heat oil in a large skillet or pot over medium-high heat. Add chicken and cook until lightly browned.
Stir in red curry paste and cook for 1 minute until fragrant.
Pour in coconut milk, fish sauce, and brown sugar. Bring to a simmer and cook for 10-12 minutes, or until chicken is cooked through and sauce has thickened slightly.
Remove from heat and stir in lime juice and fresh basil. Serve immediately with rice if desired.
Notes
Serve with jasmine rice for a complete meal. Adjust spice level by adding more or less red curry paste.