Preheat an indoor or outdoor grill over medium high heat. Grill patties, flipping once until cooked through. Add the slices of cheese to each patty and let melt for 45 seconds. Double up the patties.
Toast the buns on the grill.
On the bottom of the bun spread a layer of the garlic aioli and a dollop of the basil vinaigrette, place 2 patties on top of that followed by ribbons of red bell peppers, broken up turkey bacon and arugula. Top with the other part of the bun and serve.