In a large skillet, heat the vegetable oil over medium-high heat. Add the chicken pieces, season with salt and pepper, and cook until browned, about 5-7 minutes.
Reduce the heat to medium and add the minced garlic and grated ginger to the skillet. Sauté for 1-2 minutes until fragrant.
Pour in the coconut milk, fish sauce, and brown sugar. Stir to combine and bring to a gentle simmer.
Allow the mixture to simmer for about 10 minutes, stirring occasionally, until the chicken is cooked through and the sauce has thickened slightly.
Remove the skillet from heat and stir in the lime juice and chopped basil.
Serve the coconut basil chicken over cooked jasmine rice, garnished with additional basil if desired.