Cook the aromatics for their flavors to bloom. In a large skillet over medium, heat 2 tablespoons vegetable oil. Stir in 3 tablespoons red curry paste with 2 pressed garlic cloves, and 1 tablespoon fresh grated ginger. Cook until fragrant, about 2 minutes, stirring constantly.
Add the saucy ingredients. Add 14 ounces coconut milk, 3 tablespoons brown sugar, 2 tablespoons fish sauce, and ⅓ cup water. Stir well to combine and simmer for 5 minutes.
Add the vegetables and chicken to the sauce. Add 2 sliced carrots, 1 sliced red bell pepper, and 1 cubed sweet potato. Cook for 5 minutes until the vegetables soften. Add 1 cup cut green beans and 6 ounces thinly sliced chicken breast. Simmer until the chicken is cooked through, 3-5 minutes.
Add the final touches. Stir in the juice of ½ lime, and ½ cup torn Thai basil leaves. Taste and add more lime juice if needed. Serve with white rice and chopped red chile or jalapeño if desired.