Preheat oven to high grill. Preheat BBQ hot plate to medium (if using).
Mix all patty ingredients together and divide into 6. Roll into balls and flatten into 1.5cm thick patties.
Heat a little oil in a fry-pan on medium heat. Cook patties (in batches if needed)for 4-5 minutes each side, until browned and cooked through. Transfer to an oven tray and place on the bottom of the oven to keep warm. Reserve pan. Alternatively cook patties on BBQ.
Mix beetroot and vinegar together and season to taste. Prep tomatoes and lettuce and set aside.
Wipe same pan clean and return to medium heat with a little oil. Crack eggs into pan, season and cook for about 4 minutes, until white is set and yolk is runny. Flip and cook for a further 1 minute for hard yolk. Alternatively cook eggs on BBQ.
Cut buns in half horizontally and place on a second oven tray. While eggs cook grill buns on top oven rack for 2-3 minutes, until toasted and warmed through.
Serve everything in the middle of the table for everyone to build their own burgers.