Place 1 tbsp. sugar, 2 cloves garlic, and 1 chile on a cutting board and roughly chop everything together. Using the side of the knife, scrape the garlic–chile mixture into a rough paste. Transfer the paste to a medium bowl and whisk in remaining sugar and 1 cup warm water; continue whisking until sugar dissolves, about 30 seconds. Whisk in fish sauce and lime juice. Thinly slice remaining chile crosswise and transfer to the sauce along with the carrots. Let sauce sit until the flavors meld, at least 30 minutes. Stir well before serving.