Set up for boiling the eggs by preparing a large bowl of cold water and ice. Set aside for when the eggs are done. Bring a pot of water with 3 inches of water up to to a boil over high heat. While your water is coming to a boil, heat up a skillet or frying pan on medium-low heat and assemble your sandwich.
Butter the outsides of the bread and lay down on a cutting board, butter facing down. Evenly layer on half of the shredded cheese on one slice. Top with the sliced avocado and the remaining cheese. Top with the other slice of bread, butter side out.
By now your water should be boiling – reduce the heat until the water is barely simmering. Using a slotted spoon, gently add your eggs and set a timer for 6 minutes. Cook your sandwich slowly on medium-low heat until golden brown and toasty, flipping once. When perfect, remove from the pan and slice into soldiers. You should be just about down cooking your sandwich when the 6 minutes are up.
Use the slotted spoon to remove the eggs and immediately plunge into the ice water bath. Put the eggs into egg cups, lop the tops off, sprinkle with salt and pepper and enjoy immediately with the avocado grilled cheese soldiers.
