Chop fresh basil leaves to make ½ cup and mix them with light mayonnaise, extra-virgin olive oil, freshly squeezed lemon juice, and minced garlic. Stir the mixture until smooth and season with kosher salt to taste. Refrigerate the aioli until ready to use.

In a small saucepan, combine fresh blueberries with chili sauce, ketchup, honey, balsamic vinegar, minced garlic, kosher salt, and black pepper. Bring the mixture to a boil over medium-high heat. Lower the heat and simmer for 5 minutes until the sauce thickens slightly. Keep warm until assembling the burgers.

Preheat a gas grill to 450°F and lightly oil the grates. Form the ground beef into 4 patties, handling the meat as little as possible to avoid overpacking. Season both sides of each patty with kosher salt and freshly ground black pepper. Grill the burgers over direct heat for about 6 minutes, monitoring for flare-ups. Flip the patties and cook for an additional 4 minutes, or until they reach the desired doneness.

Spread the prepared basil aioli on both the top and bottom halves of each hamburger bun. Place arugula on the bottom bun, followed by a grilled burger patty. Spoon a tablespoon or two of the blueberry barbecue sauce over the patty, then add crumbled blue cheese and crispy bacon slices. Finish with the top bun and serve immediately.