This creamy, spicy chicken soup is reminiscent of tom ka gai but a bit heartier due to the addition of rice noodles. In lieu of lemongrass, makrut lime leaves, galangal and chiles, green curry paste is used to flavor the coconut milk–infused broth.
Preheat the oven to 450 degrees F. Place a rack in a roasting pan. Mix the butter, chopped garlic, curry paste, 3 teaspoons ginger, the zest of 2 limes and the juice of 1 lime in a bowl.
Season the chicken with salt. Loosen the skin and rub the curry butter underneath. Place skin-side up on the rack and pour about 1/2 cup water into the pan. Roast until the chicken is cooked through and the skin is crisp, 20 to 25 minutes.
Meanwhile, mix the crushed garlic with the remaining 1 teaspoon ginger, the zest of 1 lime and juice of 2 limes. Whisk in the vegetable oil. Shave the carrots into ribbons with a vegetable peeler and toss with the dressing. Add the cilantro and 1/2 teaspoon salt and toss. Serve the chicken with the carrot-ginger salad.