Portion the lamb into 4 balls, let rest at room temperature while you caramelize the onions.
Preheat the griddle to medium low heat and add the oil.
Add onion slices to the griddle and let caramelize to a golden brown, flipping occasionally, roughly 20 - 30 minutes.
Remove onions from griddle and turn to high heat.
Add the lamb to the griddle and smash down to create a nice sear, 3 - 5 seconds.
Season lamb patties with salt and pepper.
Let lamb cook undisturbed until juices run from the top, roughly 2 - 3 minutes.
Flip burgers and season top side with salt and pepper.
Add crumbled feta cheese to burgers.
Let cook for another 2 -3 minutes for medium.
Remove patties from griddle and tent with foil.
Wipe excess fat from griddle with paper towel.
Melt butter on the griddle until bubbly.
Toast buns in the butter until lightly toasted.
Remove buns from the griddle and build the burger.
Spread olive tapenade on a bottom bun, add one leaf of butter lettuce, the lamb burger, top with tzatziki sauce, onions, pickles, tomatoes and garnish with dill.
Serve while lamb is hot.