Blend chiles, garlic, sugar, fish sauce, salt, and 3 Tbsp. oil in a blender until smooth. Transfer marinade to a medium bowl and add shrimp; toss to coat. Let sit 10 minutes.
Heat remaining 1 Tbsp. oil in a large nonstick skillet over medium-high. Just when oil begins to smoke and working in batches if needed, add shrimp, leaving marinade behind, and cook, turning once, until lightly charred around the edges, about 1 minute per side. Remove pan from heat. Add basil and toss vigorously until basil is wilted.
Transfer shrimp mixture to a platter. Serve with lime wedges alongside.