Make sauce: In a small bowl, whisk ingredients until combined. Refrigerate until ready to use.
Make burger: In a large skillet over medium heat, heat 1 tablespoon oil. Add onion and cook, stirring often, until starting to soften. Reduce heat to medium-low and continue to cook, stirring often, until deep golden and jam-like, 45 to 60 minutes. Transfer to a plate and wipe skillet clean.
Add remaining 1 tablespoon oil to skillet and heat over medium-high. Cook patty 2 to 3 minutes per side, or until it reaches desired doneness.
Slather bun with sauce. Add patty, cheese, and a big spoonful of caramelized onions. Let cheese melt. Add bacon, lettuce, and tomato. Top with bun.