First things first: All you need to pull off the comfort-food staple that is an egg-topped burger is a great cut of beef and a bun that’s not too sweet. Then grill or fry your burger, and top it off with a fried egg and a funkier cheese.
"I think the gooeyness of Gruyère is a solid match for a runny yolk," says Ken Addington, executive chef at Brooklyn’s Five Leaves restaurant. But for a truly authentic dish, try Addington’s Five Leaves Burger, an Aussie-style version (after all, the Australians launched the egg-on-burger craze in the first place). "The Aussies call it a burger with ‘the lot,' which includes beet, pineapple, a fried egg, and sweet chili sauce," says Addington, who likes his eggs sunny-side up—the runnier the yolk, the better. "As soon as the bun goes on, the yolk is all over the place. It’s a fun mess."
Addington suggests starting with a medium-heat flame and cooking a fried egg more slowly than you would normally, to avoid browning. The goal is to get the white cooked solid while keeping the yolk runny.
In a medium saucepan, combine vinegar, honey, coriander, peppercorns, olive oil, mustard, and ginger. Bring to a boil, and cook for 5 minutes.
In a bowl, pour mixture over the pineapple slices, cover with plastic wrap, and refrigerate overnight. Combine harissa, mayonnaise, and lime zest and juice in a bowl; set aside.
Prepare medium-high grill or ridged grill pan. Remove pineapple from marinade, and grill for 2 minutes per side.
Season beef with salt and pepper, and form into 4 burgers. Grill the burgers (only cover if using a gas grill), turning once. Cook about six minutes for medium-rare.
Melt 2 Tbsp of butter. Lightly brush cut side of buns with it, and grill for 1 or 2 minutes. Spread harissa mayo on the grilled buns.
In a nonstick pan, melt 2 Tbsp butter over medium-low heat. Break each egg into a small bowl, and add it to the pan (see below for expert egg instructions).
Place burgers on buns, and top each with a beet slice, pineapple slice, and sunny-side up egg.
For the egg: Over medium heat, place a little butter and salt on the bottom of the pan.
Crack your egg into the pan, and season the top side of the egg with salt.
Let it do its thing for about five minutes, and do not flip over. As soon as the white is set, remove from the skillet.