In a small bowl, mix the ketchup and balsamic reduction.
In a large skillet, heat 2 tbsp. EVOO, two turns of the pan, over medium-low. Add the onions and season with salt and pepper. Cook the onions, stirring occasionally, until deep golden and very sweet, about 35 minutes. (As the onions cook, add some water to the skillet, if needed, to keep them moist.)
Preheat a cast-iron skillet or griddle over medium-high. In a medium bowl, combine the beef, wine, rosemary, sage and garlic; season. Form into 4 patties (thinner at the center for even cooking).
Drizzle the patties with some EVOO and cook, turning once, about 8 minutes for medium-rare. During the last minute of cooking, top the patties with the cheese; cover or tent with foil until the cheese melts.
Divide the burgers among the focaccia bottoms. Top each with the onions, balsamic ketchup and the focaccia tops.