Bologna barbecue rub: add all barbecue rub ingredients to a bowl and stir to combine. Taste it to see if it's super spicy and adjust.
Smoke the bologna: set up your smoker or kettle grill for cooking at 225°F (105°C) with your favorite choice of wood or pellets for pork.
With a sharp knife score the outside of the bologna. Keep your cuts about an inch apart and then make a crosshatch of scores going the other direction. Don't cut too deep into the meat.
Once the bologna has been scored, rub the outside with 3 tablespoons of yellow mustard. This mustard is going to be your "glue" to allow the barbecue rub to stick to the outside of the bologna.
Once the mustard is spread all over the bologna, clean your hands and then sprinkle the barbecue rub very well over the outside. Don't forget about the ends of the bologna. Do your best to make a consistent spread of rub.
Once your grill is set up and the internal smoker temperature is around 225 F, add the rubbed bologna and allow it to smoke for 2 to 3 hours. The bologna should take on some color, the barbecue rub should fully have adhered and after this amount of time smoking, your bologna should have taken on a lot of smoky flavors. Remove it from the smoker and allow the meat to cool.
Slice the bologna as thinly as possible. If you want to get thin slices, the meat has to be cool and if you really want to be able to slice it super thin and you don't have an electric slicer, you can put the meat in the refrigerator or even freezer for a few minutes to firm up the meat.
Assemble the sandwich: spread 1 tablespoon of butter across your slices of bread and toast them. You can do this in a toaster or in a pan with the meat if you prefer.
If your slices of bologna aren't already sliced up around the edges a little bit (from the pre-smoking scoring), add 2 to 4 small cuts around the circumference of the bologna slices. This helps it sear and stay flat in the pan.
In a medium pan or griddle over medium-high heat, add your slices of bologna. Cook for 2 minutes per side or until you start to get some browning and maybe even some darker brown spots on the meat.
Once you've started to get the meat seared, add slices or graded smoked cheese on top of the hot bologna that is still in the pan to get allow the meat to start melting the cheese. Stack all bologna in the pan with the cheese.
Add condiments to your sliced and toasted bread and get ready for the stack of bologna and cheese. Once the cheese seems like it's started melting, remove the meat and cheese to one slice of bread and get ready to cook your eggs.
Crack 2 eggs into a small bowl and with a fork, whisk the eggs until they are very whipped up and the yolks and whites are fully combined. Add a pinch of salt and pepper to the raw eggs.
Wipe out the pan or griddle and add 1 tablespoon of butter. Once the butter starts bubbling and foaming up (should be almost immediately) add your salted and peppered, mixed-up raw eggs. You might want to turn the heat in the pan off or low at this point if you see smoke or if the butter almost instantly starts to disappear.
Allow the eggs to coat the bottom of the pan and they should cook fast. Once the eggs seem to have set a little, start to fold them over themselves like a pamphlet or brochure. Try to form the eggs into the size of the bread that your sandwich will be on.
If you can't make a great folded egg, or if something goes upside down, you can easily use a spatula or knife to cut and shape the egg to fit the bread.
Add the egg to the top of the stacked bologna and cheese and then top everything with the second slice of bread.
Serve the sandwich and enjoy.