Halve and peel onion. Slice halves into thin strips. Separate leaves of lettuce. Set aside four largest pieces for burgers, and tear remaining leaves into 1" pieces. Core tomato and cut two 1/4" rounds. Cut remaining tomato into 1/2" dice. Trim cucumber, halve, and cut into 1/2" dice. Combine mayo and half the Cajun seasoning in a mixing bowl to create special sauce. Taste, and add more seasoning if desired.

Place a large non-stick pan over high heat. Add 1 tsp. olive oil and onion to hot pan. Stir constantly 3 minutes. Reduce heat to medium and stir occasionally until onions are golden brown and tender, 6-9 minutes. Season with a pinch of salt and pepper. Remove from burner. Transfer onions to a plate. Reserve pan; no need to wipe clean.

In another mixing bowl, combine ground turkey, 1/4 tsp. salt, and a pinch of pepper. Mix well and form into four equal-sized patties (about 1/3" thick and 4" in diameter). Using your hands, form blue cheese into two balls and flatten into 1" pucks. Divide cheese between two patties. Place remaining patties on cheese-topped patties and seal edges by pinching lightly. A tight seal helps cheese stay in burger while cooking.

Return pan used to caramelize onions to medium heat. Place patties in hot pan. Cook until browned, 2-3 minutes. Flip, cover, and cook until burger reaches a minimum internal temperature of 160 degrees, 2-3 minutes. Cheese may leak from burgers. Don't worry, they're still delicious!

In another mixing bowl, combine vinegar, 2 tsp. olive oil, and a pinch of salt and pepper. Toss with torn lettuce, diced tomatoes, and cucumber.

Place patty on a lettuce leaf. Spread on special sauce and top with tomato slice, caramelized onion, and a second lettuce leaf. Serve salad on the side.