Personal preference, but I prefer my burgers without breadcrumbs/eggs/onions etc in them. I love burgers, but don't enjoy the British tradition of 2 inch-thick discs of beef with a million ingredients in.
I prefer a thin patty with just non-lean beef mince and S+P once it's been formed. I'd like to try with a decent butcher mixed grind of a couple of different beef cuts, or even get a cheap grinder myself some time.
Anyway, this summer at some barbecues I've been going with:
1 beef patty, couple of cm thick and diameter just slightly bigger than the bread roll
1 Warburtons sliced white roll, toasted
1 slice of American style processed cheese slice
1 full slice of white onion, cut as thinly as possible
1 slice of beef tomato, cut as thinly as possible
1 leaf of standard non-iceberg lettuce
2 slices of gherkin
Ketchup and Mayo
Form the patties just tight enough to hold together though the cooking, but not any tighter (it takes a few tries), then season. Fry/BBQ to your preference. Once you flip the burger, put the cheese slice on the cooked side while the other side cooks. Spread mayo on the inside of one half of the bun, and ketchup on the other, then layer everything up.
Probably didn't need a recipe/process as you can suss out how to make them just by seeing one, but I do think they're better received than the standard British BBQ burger of just thick dense overcooked beef mince slathered in ketchup. As with most things, it's the little details that make it a decent burger though. Stuff like toasting the bread, properly melting the cheese and making sure the patty is the same size as the bun really make a difference. Also, don't be tempted to posh-up the ingredients too much. The cheapo warburtons rolls and processed cheese slices are top picks. The bread rolls are the only things I could find in a supermarket that even resemble the legendary martin's potato rolls that everyone in America uses. The processed cheese is there for subtle cheesiness and meltability - stuff like cheddar is too strong and doesn't melt in the right way.