Cook 1 tablespoon of the oil, the curry paste and onions in a large saute pan over medium heat, stirring often and letting sizzle, 5 to 6 minutes.
Pat the chicken dry, sprinkle with salt and pepper and add the remaining oil to the pan. Cook the chicken in the onion-curry mixture until golden on all sides. Add the broccoli, carrots, basil, garlic and lime zest and cook, stirring, until the vegetables are coated, about 2 minutes. Add the coconut milk, chicken stock and tomatoes and bring to a simmer. Let the chicken simmer until cooked through and the sauce begins to thicken, about 20 minutes. Squeeze with lime juice before serving.
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