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.