Combine all the sauce ingredients in a small bowl and mix well. Refrigerate until ready to use. Alternatively, if you want to use the sauce in a squeeze bottle, I recommend blending the ingredients so the relish doesn’t get stuck.
Heat a large cast iron skillet over medium heat and melt in 1/2 a tablespoon of butter. Place the buns in cut side down until golden brown, this will only take a minute so keep an eye on them.
Using your hands, form the ground beef into a ball and then gently flatten it into a patty that is about a 1/4” thick. Try to make it slightly larger than your bun as the patty will shrink slightly when it cooks. The exact thickness doesn’t matter but if you go any thicker or thinner, adjust your cooking time accordingly. Generously season one side with salt and pepper.
Melt the rest of the butter into the skillet and place the burger in seasoned side down (to one side so there is room for the egg). Season the other side with salt and pepper. I like my burgers medium rare and it takes about 1 and a half minutes per side to get there. Feel free to leave it on longer if you like your burger more done.
After you flip your burger, place on the cheese so it can start to melt. While your burger is cooking, drizzle some olive oil onto the other side of the pan and fry the egg until the whites are firm but the yolk is still runny.
To build your burger, spread a layer of sauce on the bottom bun and then top with sliced avocado. Place the patty on with fried egg on top. Finally, put the top bun on, roll up your sleeves, and dig in.