Thinly slice the onion. Finely chop 1 tablespoon of the onion slices. Set aside.
In a bowl combine egg, bread crumbs, blue cheese, mustard, garlic, black pepper, and the finely chopped onion. Add ground chicken breast; mix well. Shape into four ¾-inch-thick patties (mixture will be slightly soft).
In a large skillet heat 2 teaspoons of the oil over medium heat. Carefully place patties in the skillet. Cook for 14 to 18 minutes or until done (165°F), turning once halfway through cooking.
Meanwhile, in a heavy large nonstick skillet heat the remaining 2 teaspoons oil over medium-low heat. Add sliced onion to skillet; cover and cook for 13 to 15 minutes or until onion is tender. Sprinkle with sugar, if using. Cook, uncovered, over medium-high heat for 4 to 5 minutes more or until onion is golden brown, stirring frequently.
To serve, top bottom half of each bun with the patties. Spoon caramelized onions over patties. Top with salad dressing, tomato slices, lettuce leaves, and bun tops.
Serve with Homemade Kale Chips and Easy Homemade Shandy (see associated recipes).