In a large bowl, mix together beef, Worcestershire sauce, egg, bread crumbs, salt, black pepper, soy sauce and olive oil until combined.
Form into 4 equal-size patties, making sure not to compact them too hard, or they will become dense and heavy. Refrigerate until the caramelized onions are finished.
For the caramelized onions, heat olive oil over medium-high in a medium saucepan. Add onions and cook for 5 minutes. Add a pinch of salt, ¼ cup (60 mL) water, balsamic vinegar and sugar. Lower heat to medium-low, cover and cook for 30 minutes. Remove lid, continue cooking to thicken the liquid and glaze the onions, about 3 minutes. Set aside.
When onions are halfway finished, preheat grill to medium-high.
Grill the burgers on one side until grill marks are attained, about 5 minutes with lid closed. Flip and grill the second side another 5 minutes. If burgers need more time to cook, move them to a cooler area on the grill to avoid burning. Finish cooking to desired doneness.
Fill each bun with a piece of romaine lettuce, then a burger, top with onions and a large piece of blue cheese.