In a medium skillet, melt butter over medium-low heat.

Add the sliced onions and cook until they are caramelized, which should take about 15-20 minutes. Stir often and add water if needed to avoid burning. Once caramelized, remove from skillet and tent to keep warm.

In a large bowl, mix together ground beef, garlic powder, salt, and pepper. Shape the mixture into 8 square patties, slightly larger than the bread and about ¼ inch thick.

Clean the skillet and set the heat to medium-high. Cook the hamburger patties in batches (two at a time) for 1-2 minutes per side. Once cooked, remove the patties from the skillet and set aside.

Toast 4 pieces of Texas toast and set them aside.

In a small bowl, combine ⅓ cup mayonnaise, lemon juice, pepper, garlic, parsley, and ketchup to make the aioli spread.

To assemble the sandwich, start with 4 slices of untoasted bread. Top each with a layer of cheese, a hamburger patty, caramelized onions, and then each piece of toast with aioli spread on both sides. Add more cheese and a second beef patty, more onions, and finish with a final slice of bread. Spread 1 teaspoon of mayonnaise on top.

Clean the skillet again and set it to medium heat. Carefully add the sandwiches to the skillet mayo-side down. Place a weighted pan on top to press the sandwich firmly and cook until golden brown, about 3 minutes. Remove the weight, spread 1 teaspoon of mayonnaise on the top slice, and gently flip the sandwich. Cook for an additional 3 minutes or until golden brown.

Serve warm.