Prepare the Ingredients: Slice the chicken into bite-sized pieces. Peel the skin off the kabocha squash and cut into 1" square cubes. Take the vein out of the kaffir lime leaves, and tear each leaf in half. Remove the Thai basil leaves from the stem.
Fry the Curry Paste: Open your can of coconut milk and scoop off the top ⅓ cup to get the thicker coconut cream. Heat the coconut cream over medium high heat until it starts to bubble and separate (you will see small bubbles that look oily). Add your red curry paste and fry in the coconut cream until you can really smell the paste (about 5 minutes).
Cook the Curry: Add your chicken, coat with the curry coconut paste, and let cook for a minute. Then add the other ⅔ cup of thinner coconut milk and 1 cup water and let simmer. Add in the kabocha squash and torn kaffir lime leaves and allow to simmer until the kabocha squash is cooked. You may need to add more water, depending on how long you allow the squash to simmer.
Adjust the Seasonings: While the squash is cooking, taste the curry and adjust the flavors. I generally need to add just a teaspoon each of fish sauce and sugar, but this will depend on which curry paste you use (see Note). Once your curry is seasoned to your liking and the squash is cooked, add your Thai basil leaves.
Serve the Curry: Allow the Thai basil to cook for a minute or so, then remove the pot from the heat. Garnish with a sprig of fresh Thai basil. Serve the curry with warm jasmine rice.
Video
Notes
My preferred curry paste brand is Mae Anong. For this brand, I tend to need 1 teaspoon each of fish sauce and sugar to round out the curry. Other curry paste brands will have different flavor profiles and require different amounts of fish sauce and/or sugar.