Heat half of the avocado oil in a skillet over medium-low heat.
Add the thinly sliced yellow onions to the skillet and cook, stirring occasionally, for about 15 minutes until they are soft and golden brown.
Deglaze the pan by stirring in the balsamic vinegar, cook for 1 more minute, then remove the onions and set aside.
In a bowl, gently season the ground beef with sea salt and black pepper, then form into a single 1-inch thick patty.
Wipe the skillet clean, increase heat to medium-high, and add the remaining avocado oil.
Place the beef patty in the hot skillet and sear for 4-5 minutes per side, or until it reaches your desired level of doneness.
While the burger rests, crack the egg into a small non-stick pan and fry until the whites are set but the yolk is still runny.
Lightly toast the sprouted grain bun in the skillet or a toaster until golden.
Assemble the burger by placing the baby arugula on the bottom bun, followed by the beef patty, caramelized onions, and the fried egg.