Season the steak all over with 1-1/2 teaspoons salt and 1/4 teaspoon pepper. Heat 1 tablespoon of the oil in a large heavy-duty skillet over medium-high heat. Cook the steak, flipping once, until well browned outside and medium rare inside, 10 to 12 minutes total.

Meanwhile, put the remaining 1 tablespoon oil, 2 tablespoons of the lime juice, 1-1/2 tablespoons of the fish sauce, and 1 teaspoon of the sugar in a large bowl and whisk to combine and dissolve the sugar. Add the carrots, cilantro, and chiles and toss well to coat.

When the steak is done, transfer to a large plate, loosely cover with foil, and keep warm. Return the skillet to the stovetop over medium-low heat. Add 1/3 cup water and bring to a boil, scraping with a wooden spoon to release any brown bits. Add the coconut milk, curry paste, and the remaining 1 tablespoon lime juice, 1/2 tablespoon fish sauce, and 1 teaspoon sugar; cook, whisking constantly, until thickened and fragrant, 4 to 5 minutes. Season to taste with salt and pepper.

Thinly slice the steak across the grain and transfer to plates. Spoon the sauce over the top and serve with the carrot salad on the side.