Fry the curry paste in coconut cream: Open the coconut milk without shaking it and spoon off the thick cream that rises to the top into a large skillet or Dutch oven. Heat the coconut oil over medium heat, add the thick coconut cream, then stir in the yellow curry paste. Fry for 2-3 minutes, stirring constantly, until the paste is deeply fragrant and the oil begins to separate.
Add the chicken: Add the chicken pieces to the pan and toss to coat every piece in the fried curry paste. Cook for 3-4 minutes, stirring occasionally, until the outsides are no longer pink. The chicken doesn't need to cook through yet — it will finish simmering in the sauce.
Add coconut milk, broth, and vegetables: Pour in the remaining coconut milk and the chicken broth, scraping up any paste stuck to the bottom of the pan. Add the smashed lemongrass, torn kaffir lime leaves, potato chunks, carrots, and onion wedges. Stir to combine.
Simmer until tender: Bring to a gentle boil, then reduce the heat to maintain a steady simmer. Cook uncovered for 12-15 minutes, stirring occasionally, until the potatoes are fork-tender and the chicken reaches 165°F internally. The sauce should be creamy and lightly thickened.
Season and finish: Fish out and discard the lemongrass stalk and kaffir lime leaves. Stir in the fish sauce, brown sugar, and lime juice. Taste and adjust: more fish sauce for saltiness, a pinch more sugar to balance the heat, a squeeze more lime for brightness. Stir in the Thai basil leaves and let them wilt for 30 seconds. Serve hot over steamed jasmine rice, garnished with extra basil.