Caramelize the onions: Heat butter in a large skillet over medium heat. Add sliced onions, salt, sugar, and thyme. Cook, stirring often, for 25–30 minutes until deeply golden. Add balsamic vinegar in the last few minutes, if using. Set aside.

Make the spread: In a small bowl, mix mayo, Dijon mustard, and lemon juice until smooth. Refrigerate until ready to use.

Cook the burgers: Shape ground beef into 4 patties. Season both sides with salt and pepper. In a hot skillet or grill pan, cook patties for 3–4 minutes per side (for medium), adding a slice of cheese on top during the last minute to melt.

Assemble the burgers: Spread Dijon mayo on toasted buns. Place a patty on each bun, then top with a generous pile of caramelized onions. Add the top bun and serve hot.