Prepare your chicken and vegetables. Heat a large pan over medium-high heat. Open coconut milk and scrape the congealed coconut oil into the pan. (If there is no coconut oil on top, simply add about 1/2 cup of the coconut milk.)
Add curry paste and let cook for a minute or two, stir, then on one side of the pan add chicken and on the other the vegetables. Stir each pile to coat in the curry and spread out, allowing to cook for a few minutes.
As the chicken begins to turn white on all sides and the vegetables start to become a bit tender, add the coconut milk, sugar, and fish sauce and simmer until the chicken is cooked through and the sauce is of the desired consistency.
Since Thai curries are traditionally rather thin and soupy, this takes just minutes. If too thick, add a little water.
Serve with rice and lime.
Note: For a thick curry, either: