In a medium skillet, melt butter over medium-low heat. Add onions and cook until caramelized (about 15-20 minutes), stirring often. Add water as needed to prevent burning. Remove from skillet and tent to keep warm.
In a large bowl, mix together ground beef, Worchestershire sauce, garlic powder, salt, and pepper.
Make 8 square patties (a little larger than the bread). Each patty should be about ¼ inch thick.
Wipe out the skillet and set the heat to medium-high. Add the hamburger patties in batches (two at a time) and cook 1-2 minutes per side. Remove patties from the skillet and set aside.
Toast 4 pieces of the Texas toast and set aside.
Meanwhile, in a small bowl, combine the mayonnaise, lemon juice, pepper, garlic, parsley, and ketchup to make the aioli spread.
Top 4 untoasted slices of bread with cheese, hamburger patty, onions, toast that has aioli spread on both sides, more cheese, a second beef patty, more onions, and top with a slice of bread.
Spread 1 teaspoon of mayonnaise on the top.
Wipe out skillet and set heat to medium. Add sandwiches to the pan, mayo side down. Place a weighted pan on top to hold the sandwich firmly in the pan. Cook until golden brown (about 3 minutes). Remove the weighted pan. Spread 1 teaspoon of mayo on the top slice of bread and flip carefully. Cook for 3 additional minutes (or until golden brown). Serve warm.