The secret to impossibly crisp skin and juicy meat? Coating the wings in cornstarch and frying twice, first at a low temperature and again at a high temperature. You can make the accompanying blue cheese dipping sauce a day in advance.
Blue Cheese Dip: In a bowl, stir together sour cream, mayonnaise, and lemon juice. Fold in blue cheese; season with salt and pepper. Cover and refrigerate until ready to serve, up to 1 day ahead.
Wings: Pour 2 inches oil into a large, heavy-bottomed pot (to ensure safe frying, pot should not be more than one-third full). Heat over medium-high until a deep-fry thermometer reads 320 degrees. Reduce heat to medium to maintain temperature. Meanwhile, combine cornstarch, 2 teaspoons salt, and 1/4 teaspoon pepper in a large resealable plastic or brown paper bag. Add wings, seal or fold bag closed, and shake vigorously to evenly coat.
With a Japanese spider or slotted spoon, add one-third of wings to oil, stirring a few times, until cooked through and oil returns to 320 degrees, 6 to 8 minutes. Transfer to a wire rack set on a rimmed baking sheet. Fry remaining wings in two batches. Let cool completely, 15 minutes. Meanwhile, combine butter, garlic, and cayenne in a saucepan. Bring to a simmer over medium-high heat, stirring until butter melts; simmer 1 minute more. Remove from heat; discard garlic. Stir in hot sauce.
Increase heat to medium-high and bring oil in pot to 380 degrees. Return one-third of wings to oil, stirring occasionally, 5 to 7 minutes. Return to rack-lined baking sheet; let excess oil drain to prevent sogginess, 30 seconds. In a large bowl, toss with one-third of hot-sauce mixture; serve immediately with dip and vegetable sticks. Fry remaining wings in two batches, returning oil to 380 degrees in between. To serve all batches at once, keep wings warm on rack in a 225-degree oven, and toss in sauce all together at end.