Heat oil in skillet. Mix meat, worcestershire, water and seasonings. Mix well with hands. Make patties (they will be a bit "runny" because of the water, but that's ok.) Put patties in skillet, add diced onion and scatter around the burgers. Stir onions from time to time so as not to burn. Cook burgers until desired doneness - 3-4 minutes per side is good; flip burgers and continue to cook. Add cheese to top of burgers. When onions are tender, add to top of cheese. Remove and place on rye bread. Brush sides of rye bread with butter and add back to the skillet to toast each side of bread.