Start by melting butter in a skillet over medium-low heat. Add the thinly sliced onions and stir often, letting them cook slowly for about 15–20 minutes. Add small amounts of water if they begin to stick. Once caramelized, remove the onions from heat and cover to keep warm.
In a large bowl, combine ground beef, ketchup, garlic powder, salt, and pepper. Mix gently—overmixing can make the patties tough. Shape into 8 square patties slightly larger than your bread. Heat a skillet over medium-high and cook each patty for 1–2 minutes per side. Set them aside.
Mix mayo, lemon juice, pepper, minced garlic, parsley, and ketchup in a small bowl. Stir well until smooth. This creamy spread adds big flavor and a hint of zing.
Lightly toast 4 slices of the Texas toast and set them aside. These will go in the center of the sandwich stack.
On 4 untoasted slices of bread, layer cheese, one cooked beef patty, caramelized onions, a toasted bread slice with aioli on both sides, more cheese, a second beef patty, extra onions, and top with another slice of bread. Spread mayo on the top slice.
Heat the skillet again and place each sandwich mayo-side down. Place a heavy pan or press on top and grill for about 3 minutes. Remove the weight, spread mayo on the top slice, and carefully flip. Cook the other side for another 3 minutes until golden brown and crispy.