Burgers:
Make six hamburger patties by shaping with your hands into 1/2-inch-thick patties about 4 1/2 inches in diameter . Season well on both sides with a seasoning blend.
Preheat the grill to medium-high heat. Transfer hamburger patties to the hot grill grates and grill for 3-4 minutes on both sides for burgers that are medium-rare to medium. If you prefer your burgers to be closer to medium to medium-well done, cook them for 4-5 minutes on each side. Add cheese during the last minute or two of grilling so it can melt.
Remove cooked burgers from grill and let rest for 5-10 minutes.
Blueberry Compote:
In a small saucepan, combine the blueberries, sugar, balsamic vinegar, lemon juice and salt over medium-high heat. Bring to a broil, then reduce heat to simmer and cook, stirring frequently for 10 minutes, until berries have burst.
In a small separate bowl, combine the water and cornstarch to make a slurry, then stir into the blueberry compote. Continue to stir until the mixture thickens slightly, then remove from heat and set aside to cool.
Basil Aioli:
In a food processor or blender, process the basil, lemon juice, garlic, and oil until smooth, then add the mayonnaise and blend or stir to combine.
Assembly:
To assemble the burgers, arrange each burger on the bottom bun, then top with two slices of cooked bacon, red onion, blueberry compote, and arugula. Spread the cut side of the top bun with basil aioli, then place on top of the burger. Serve immediately.