In a large skillet over medium–high heat on the stove, warm the oil. Add the onion and cook for 3 minutes, stirring occasionally, until lightly browned. Reduce the heat to medium–low and cook another 5 minutes, stirring occasionally, or until well softened. Add the wine, thyme, ½ teaspoon of salt and ¼ teaspoon of pepper and bring to a simmer. Reduce the heat, simmer until the liquid reduces to a glaze and the onions are very tender, about 10 minutes, stirring often. Remove from the heat and cover to keep warm.
Prepare the grill for direct cooking over medium–high heat (200ºC to 260ºC).
In a medium bowl mix the ground chuck, paprika, ½ teaspoon of salt and ½ teaspoon of pepper. Gently form four patties of equal size, each about 2 centimetres thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 centimetres wide in the centre of each patty to prevent them from doming as they cook. Refrigerate the patties until ready to grill.
Grill the patties over direct medium–high heat, with the lid closed, until cooked to medium doneness, 9 to 11 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt and toast the buns, cut-side down, over direct heat.
Build each burger on a bun with a patty, caramelized onions and baby greens. Serve right away.