In a large skillet or wok, heat 3 tablespoons of oil over medium heat. Add the chopped leek, minced ginger, and minced garlic. Sauté for about 2-3 minutes, or until the leek is soft and fragrant.
Stir in the chopped yam, cabbage, green bell pepper, and snap peas. Cook for another 5-7 minutes, stirring occasionally, until the vegetables begin to soften.
Add 1-6 tablespoons of red curry paste to the vegetable mixture, depending on your desired spice level. Stir well to coat the vegetables evenly with the paste.
Slowly add 1 ½ cups of slow-sodium chicken broth and the entire can of unsweetened coconut milk. Bring the mixture to a gentle simmer.
If using chicken, add the chopped boneless skinless chicken breast to the curry. If opting for shrimp, wait until the curry is simmering before adding the shrimp. Cook for about 5-7 minutes or until the chicken is cooked through or the shrimp is pink and opaque.
If you prefer a thicker curry, mix 1 or 2 teaspoons of cornstarch with a tablespoon of water to create a slurry. Stir this into the curry and let it cook for an additional 2 minutes.
Stir in the fish sauce, lime juice, and light brown sugar. Taste and adjust seasoning, adding more lime juice or sugar as needed.
Remove the curry from heat, and fold in the Thai basil leaves. Serve hot over jasmine or white rice if desired, and garnish with fresh cilantro and thinly sliced shallots.
