Prepare the eggplant: Slice the eggplant into ½ inch thick rounds. Arrange them on a paper towel-lined baking sheet and sprinkle generously with salt. Let them sit for 30 minutes to draw out excess moisture. Pat them dry with paper towels.
Set up the breading station: In three separate shallow bowls, whisk the flour with salt and pepper in one, the beaten eggs in the second, and the panko breadcrumbs in the third.
Bread the eggplant: Dredge each eggplant slice in the flour, then coat it completely in the egg mixture. Finally, press both sides firmly into the panko breadcrumbs to ensure good coverage.
Fry the eggplant: Heat enough avocado oil in a large skillet over medium heat to reach about 350°F (175°C). Carefully fry the breaded eggplant slices for 2-3 minutes per side, or until golden brown and crispy. Drain on paper towels to remove excess oil.
Make the garlic aioli: In a small bowl, whisk together the minced garlic, mayonnaise, lemon juice, salt, and dried parsley.
Assemble the sandwiches: Spread a generous amount of garlic aioli on each toasted bread slice. Layer with lettuce, tomato slices, red onion slices, cucumber slices, American cheese, and a crispy eggplant slice. Top with the other bread slice and enjoy!