Thinly slice the yellow onion into half-moons.
Heat a small non-stick skillet over medium-low heat with a tiny drop of oil and cook the onions, stirring occasionally, until they are soft and caramelized to a deep brown, about 15 minutes.
Divide the ground beef into two 3-ounce portions and gently roll them into balls without overworking the meat.
Heat a cast-iron skillet or heavy griddle over high heat with the avocado oil until it begins to shimmer.
Place the beef balls into the hot skillet and immediately press down firmly with a heavy metal spatula until the patties are very thin and wide.
Season the patties with sea salt and black pepper; cook undisturbed for 2 minutes until the edges are dark and crispy.
Flip the patties, place the cheddar cheese slice on one patty, and cook for an additional 60 seconds.
Lightly toast the sprouted grain bun in the skillet or a toaster.
Spread the yellow mustard on the bottom bun, then layer with the butter lettuce, tomato slice, both beef patties, and the caramelized onions before closing the burger.