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In a large frying pan or wok over medium heat, let oil warm up. Add the onion and cook until slightly softened, about 3 minutes. Add any crunchy vegetables that will need longer cooking than others (I usually add the carrots at this point and hold off on the beans and squash for a couple of minutes).
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Add the chicken and salt and pepper the whole mix. Cook for about 2-3 minutes, until the chicken is cooked on the outside.
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Stir the curry paste into the pan and cook for about 30 seconds. Add coconut milk, broth, and fish sauce.
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Bring to a simmer, stirring occasionally, scraping up the browned bits on the pan bottom, cooking until vegetables are as soft as you'd like them to be and chicken is cooked through.
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If you want it a little thicker (I go back and forth on whether or not I do this), mix the cornstarch and water together until smooth and pour in about half of it, stirring constantly. After about thirty seconds, it should have thickened up a little. Use the rest of the cornstarch if you still want it thicker.
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Add lime juice and sprinkle basil over the top.
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Serve over rice.