Make the onions: Heat oil in a medium skillet over medium-high heat. Add onions and cook, stirring occasionally, until soft, about 8 minutes. Reduce heat to medium-low and continue cooking, stirring occasionally, until nicely browned and caramelized, about 30 minutes more.
Add wine, thyme, and 1/4 cup water. Raise heat to medium-high and cook, stirring occasionally, until the liquid has evaporated, about 3 minutes. Season with salt and pepper.
Make the burgers: heat oil in a small skillet over medium-high heat. Add onion and cook, stirring occasionally, until translucent, about 2 minutes. Remove from heat and let cool completely.
Heat a grill to medium-high. Combine beef, parsley, and onion in a bowl. Season with salt and pepper. Form into 8 one-inch-thick patties. Grill 4 to 5 minutes per side for medium-rare. Top with a square of cheese after flipping burger if desired. Assemble with caramelized onions and skewers.