Heat a Dutch oven, wok, or saute pan over medium heat. Add ¾ cup coconut milk (if using canned coconut milk that has separated, use the top, thicker portion of the milk) to the pan. Simmer coconut milk, stirring constantly, until it has reduced by half and you can see oil separating from the milk, 5 to 6 minutes. (Note: This step to “break” the coconut milk is part of the traditional process, but don’t worry if your coconut milk doesn’t show signs of separating after this initial cook time - just proceed with the recipe.)
Add curry paste and saute until fragrant, 2 to 3 minutes.
Add chicken breast and stir to coat in the curry paste.
Pour chicken stock and 1 cup coconut milk over chicken.
Stir in squash, bell pepper, palm sugar, and fish sauce. Bring to a simmer and let curry simmer gently until chicken is cooked through, 10 to 15 minutes. (Reduce heat as needed to prevent the curry from reaching a hard boil.)
Taste the curry and adjust the flavor, adding more sugar or fish sauce if needed.
Remove curry from heat and stir in Thai Basil leaves.
Serve curry over rice.