Heat garlic oil in medium saucepan over medium heat until shimmering. Add chiles, ginger, and curry powder and cook until fragrant, about 30 seconds. Stir in ½ cup water, peanut butter, 3 tablespoons vinegar, soy sauce, and sugar and bring to simmer. Cook, stirring occasionally, until thickened slightly and flavors meld, about 2 minutes. Adjust consistency as needed with additional water; set peanut sauce aside.

Combine carrots and remaining 2 tablespoons vinegar in small bowl; set aside. Bring 4 quarts water to boil in large pot. Remove from heat, add noodles, and let sit, stirring occasionally, until soft and pliable, but not fully tender. Drain noodles and rinse under cold running water until chilled. Drain noodles well again and toss with 2 teaspoons garlic oil; set aside.

Heat 1 tablespoon canola oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add edamame and cook until spotty brown but still bright green, about 2 minutes; transfer to bowl. Heat remaining 4 teaspoons oil in now-empty skillet over medium heat until shimmering. Add noodles, 1¼ cups water, and ½ cup peanut sauce and cook until sauce has thickened slightly and noodles are well coated and tender, about 1 minute.

Divide noodles among individual serving bowls, then top with carrots, edamame, and cabbage. Drizzle with remaining peanut sauce, sprinkle with peanuts and basil, and serve.