Pat duck breasts completely dry with paper towels. Use a sharp knife to score the skin in a crosshatch pattern, cutting just through the fat without piercing the meat. Season both sides generously with salt. Let sit at room temperature while you prep the remaining ingredients.

Place duck breasts skin-side down in a cold large skillet. Turn heat to medium-low and cook undisturbed for 12-14 minutes, pouring off excess fat as it renders, until the skin is deep golden and crispy. Flip and cook 3-4 minutes more for medium-rare (130F internal). Transfer to a plate, tent with foil, and rest 10 minutes. Pour off all but 1 tablespoon fat from the pan.

Return the skillet to medium heat. Add the red curry paste and fry for 1-2 minutes, stirring constantly, until fragrant and the oil begins to separate and sizzle at the edges.

Pour in the coconut milk and chicken stock, scraping up any browned bits. Add the palm sugar and fish sauce. Bring to a gentle simmer and cook 8-10 minutes, stirring occasionally, until the sauce thickens enough to coat the back of a spoon.

Stir in the Thai basil and sliced chile if using. Slice the duck breasts crosswise into 1/4-inch pieces. Arrange over steamed jasmine rice, spoon the curry sauce generously over top, and garnish with extra basil leaves and lime wedges.