Whisk together the mayonnaise, sriracha, rice vinegar, sugar, and lime juice in a small bowl until smooth and well combined—this is your bang bang sauce base. In a separate bowl, toss the shredded cabbage with 3 of the green onions (sliced), then set aside. Slice the remaining 3 green onions and reserve for garnish. I like to make the sauce and prep the vegetables first so everything is ready when the shrimp finishes cooking.
Pat the shrimp pieces dry with paper towels—this is crucial for getting a good sear. Heat 2 tablespoons of oil in a large skillet over medium-high heat until shimmering, then add the shrimp in a single layer. Cook for 1-2 minutes per side until the shrimp turns opaque and pink, then remove from heat. The shrimp will continue cooking slightly as it cools, so avoid overcooking. Toss the cooked shrimp with the garlic powder while still warm.
Lay out 4 tortillas on a clean surface. Divide the pepper jack cheese evenly among them, spreading about 3 tablespoons on each tortilla. Top each with a portion of cooked shrimp from Step 2, then add a generous handful of the cabbage slaw from Step 1. Drizzle with 2 tablespoons of the bang bang sauce from Step 1, then top each with another 3 tablespoons of cheese. I find that reserving some sauce for drizzling at the end gives you more control over how saucy each bite is. Place the remaining 4 tortillas on top to form quesadillas, pressing gently to secure.
Heat 1 tablespoon of oil in a large skillet over medium heat. Once shimmering, carefully place one quesadilla in the pan and cook for 3-4 minutes until the bottom is golden brown and crispy. Flip carefully and cook the second side for 1-2 minutes until golden. Transfer to a cutting board and repeat with remaining quesadillas, adding more oil to the pan as needed between batches.
Using a sharp knife or pizza cutter, slice each quesadilla into 4 triangular pieces. Arrange on serving plates and drizzle with the remaining bang bang sauce from Step 1. Garnish generously with the reserved sliced green onions. Serve immediately while the quesadillas are still warm and crispy.