Using a very sharp blade, draw a knife down the length of 1 snapper fillet to make parallel ¼" deep cuts spaced ½" apart from end to end. Cutting on a diagonal (or crosswise), make parallel cuts to create ½" diamonds or squares in a crosshatch pattern. Repeat with remaining snapper fillets.
Season both sides of snapper with salt. Place on a rimmed baking sheet. Combine garlic, fish sauce, lime juice, chili paste, and brown sugar in a small bowl; spoon over snapper, making sure marinade penetrates score marks.
Heat broiler. Drizzle snapper with oil and broil until charred in spots and cooked through, about 5 minutes. Serve with lime wedges for squeezing over.