Biscuit dough: weigh your flour. Grate frozen butter into the sifted self-rising flour. Stir to combine until the butter is fully coated with flour.
Put the bowl in the freezer for 10 minutes. It's important to keep the butter as cold as possible through this process.
Combine buttermilk with the cold flour/butter to bring it together as dough. Stir no more than 15 times. Overworking the dough at this stage could cause your finished biscuits to be tougher. Your dough at the end of 15 stirs will be very shaggy and not smooth at all.
On a floured surface dump out dough and roll with a rolling pin. If you don't have a rolling pin, you can just use your hands. Dust with flour as needed when the dough gets sticky. Fold the dough over on itself five different times. This folding process is what creates the layers in the biscuits.
Roll or flatten the dough to 1/4 to 1/2 inch thick/tall. Using a 3-inch biscuit or cookie cutter, cut out circles of dough. Do not twist the biscuit cutter, cut straight down.
Place all cut-out biscuits on a plate or sheet pan, cover with plastic wrap, and place in the refrigerator while you prepare the onions.
Preheat oven to 450 F (230 C).
Cheesy balsamic onions: add sliced onions to a small bowl and top with olive oil and balsamic vinegar. Season the onions with just a pinch of salt and pepper. Toss the onions to combine everything.
On a parchment-lined sheet pan, place small mounds of onion about the diameter of your biscuits (3.5 inches). Top with some cheese and then lay a cut-out biscuit on top of the cheese. Feel free to sprinkle a bit of black pepper on the top of the biscuit at this point if you want.
Bake the pan of cheesy, onion biscuits for around 15 minutes. Check around the 12-minute mark to make sure they do not brown too much.
Remove biscuits from the pan to cool.
Note: the following instructions are for one sandwich, but you can easily double or quadruple it if you are making more than one.
Tater tot patty: cut a small piece of parchment or wax paper to about 5 inches square. Add a little oil to the top or spray it with non-stick spray.
Add a whole egg to a bowl and whisk. A tiny bit of this egg will go in your tater tot patty and the rest will be the egg in the sandwich. SAVE THIS EGG.
If your tater tots are still frozen, you can microwave them for 45 seconds to 1 minute until they are soft enough to use. Once they are thawed, sprinkle them with a little bit of salt and black pepper and mash them with a fork and form them into a patty on top of the piece of parchment. If you have a biscuit cutter that is around 3.5 to 4 inches in diameter, you can use that for a consistent size. Otherwise, form into a patty and then leave on the parchment until ready to cook.
In a large skillet, over medium heat add 2 teaspoons of oil. Once the oil is shimmering, carefully add your tater tot patty. Use the piece of parchment to carefully place the potato patty into the hot pan on top of the oil. Cook the tater tot patty for 6 minutes on the first side before flipping.
Once the tater tot patty has cooked for 5 minutes, carefully flip and continue to cook for an additional 4 minutes to fully crisp it up. Once that time is complete, remove the tot patty to a plate while you cook the egg.
Folded egg: if you're using the same pan that you cooked the steak and onions in, remove it from the heat and wipe it out really well to remove any cooked bits. Grab the bowl with the egg in it from earlier, add salt and pepper to the egg to season. Add the pan (or a new pan) back to medium-low heat and add a tablespoon of butter.
Once the butter is bubbly and melted, pour your whisked egg on top of the butter. Immediately start letting the egg spread out on the bottom of the pan so that it all starts to cook. While it is cooking, you can help it out by keeping the egg in a round or square-ish shape and tilting the pan to allow the runny egg on top to roll off and touch the pan itself. Try to keep the shape somewhat consistent. This will help with folding.
Once most of the egg is set (after 1 to 2 minutes) you can start to fold. Using a spatula, attempt to scoop up one side and fold it over the other. This isn't always easy to do, and it doesn't really matter if you mess it up. But you effectively want to fold everything over itself once or twice until the egg is folded into a size that is consistent with your sandwich.
Once the egg is folded, remove it to the plate with your tater tot patty and assemble your sandwich.
Sandwich assembly: grab one biscuit and turn it upside down so the cheese and onion are facing up. Add slices of avocado and lightly salt the avocado.
Top the avocado with the tater tot patty and then the folded egg. Use a second biscuit with cheese and onion facing downward as the top and complete the sandwich.