Heat a large saucepan or cast-iron skillet over medium heat. Use a knife to score the duck skin five or six times to cut through the layer of fat, then add the duck to the pan, skin side down. Cook without moving the duck until the skin is golden and crisp, about 7 minutes. Remove the duck and set aside.
Pour out the remaining fat and reserve for later (it’s great for stir-frying), leaving about 3 tablespoons in the pan. Stir the curry paste into the duck fat and cook until fragrant, less than 1 minute. Add the coconut milk, broth, and squash, stirring to make sure the squash is submerged. Bring to a boil, then reduce to a simmer, cover, and cook, stirring occasionally, until the squash is tender, 20 to 25 minutes. Meanwhile, slice the duck crosswise against the grain ½ inch thick.
Once the squash has been cooking for 10 to 15 minutes, uncover the pan and stir in the sliced duck, brown sugar, and fish sauce. Continue cooking, uncovered, until the duck is firm but slightly springy, about 5 minutes. Remove the pan from the heat and gently stir in the basil and red chiles. Serve with rice or roti bread.
