Prepare all your ingredients before you begin. In bowl, beat egg and water with fork; mix in bread crumbs, onion, mustard, salt, Worcestershire and pepper.
Mix with your hands until evenly combined. Shape into four 1/2-inch (1 cm) thick patties.
Place the patties onto a tray lined with- grease proof paper. Cover with plastic wrap and place in the fridge for at least 30 minutes to rest.
Chilling the patties will help them hold together when cooked. This also allows the flavours in the patties to blend and develop. If you chill the patties for any longer, increase the cooking time slightly.
( For Make-ahead Beef Patties : Cover and refrigerate for up to 6 hours. Or layer between waxed paper in airtight container and freeze for up to 1 month; thaw in refrigerator.)
Heat half the olive oil in a large, non-stick frying pan over medium-low heat and cook half the patties for about 4 minutes on each side or until browned and cooked through.
Transfer the patties to a plate, set aside and keep warm. Repeat with the remaining oil and patties.
Meanwhile, preheat grill on high. Place the hamburger buns, cut-side up, under the preheated grill and toast for 1 minute or until golden. Leave the grill on.
Top the bottom halves of the toasted hamburger buns with the patties and then the cheese slices. Return the bottom bun halves under the grill for 1 minute or until the cheese is just melted.
Top each with the sliced tomato, tomato sauce and shredded lettuce, tomato, cucumber and onion rings, sprinkle with salt and pepper and then cover with the hamburger bun tops.