Preheat the oven to 350 degrees F.
Open the tuna with a can opener, then drain off any excess water. Dump the tuna into a medium-sized bowl, using a fork to get remaining bits out if necessary.
Break up any large tuna pieces with the fork, then measure out the half cup of mayonnaise with a measuring cup.
Mix the mayonnaise into the tuna until both are well combined. Set aside.
Toast the individual slices of bread in a toaster, until they start to brown slightly. This will help ensure the bread doesn’t get soggy
Once bread is toasted, place evenly on a sheet pan, with several inches between each slice.
While the bread is toasting, prepare the avocado and tomato.
Using a cutting board and small sharp knife, cut the avocado in half. Take the pit out of the other half of the avocado (carefully!) with a spoon or the knife. Or for a safer method, see a tutorial here.
Cut the avocado into roughly ⅛ inch thin slices with the knife. You will be cutting it in it’s peel, being careful not to pierce the skin. Set aside.
Wash the tomato and place it on the cutting board. Use a very sharp knife to slice up the tomato. First cut a quarter-inch slice off the top, then seed it by scooping out the seeds with a spoon. Then slice the rest of the tomato and set aside.
Place the tuna mixture onto each slice of bread with the fork, dividing it evenly between the four pieces and spreading it out into an even layer.
Scoop the avocado slices out with a spoon and place them in a single layer on top of each piece of toast over the tuna mixture.
Place one slice of tomato on top of each toast, then top with a slice of cheese.
Put the sheet pan in the oven and set the timer for 15 minutes.
While the tuna melts are in the oven, place a medium pan over medium heat. Once it’s warmed, add in half the butter and let it melt.
After the butter is melted, twist the pan so that the butter covers the entire bottom. Then crack one egg into the pan without breaking the yolk. If your pan is big enough, you can do two eggs at a time, just make sure the eggs don’t touch otherwise they’ll meld together and be difficult to get out of the pan.
Place a lid over the pan and cook the egg(s) until the whites are done but the yolks are still soft, roughly 4 minutes. You can cook the egg(s) longer if you prefer a hard yolk.
After the first two eggs are done cooking, use a spatula to carefully place them on a plate and set them aside. Then put the remaining butter in the pan and repeat the process for the remaining two eggs.
When the timer goes off, remove the sheet pan from the oven and place an egg on top of each gluten free tuna melt, then serve!