In a large bowl, gently mix the ground beef with garlic powder, onion powder, salt, and pepper. Do not overmix.
Divide the beef mixture into 4 equal portions and form them into patties, about 3/4 inch thick. Make a slight indentation in the center of each patty with your thumb to prevent them from puffing up during cooking.
Heat a large skillet or griddle over medium-high heat. If not using a non-stick skillet, add a tablespoon of oil.
Place the patties on the hot skillet. Cook for about 3-4 minutes per side for medium-rare, or longer depending on your desired doneness. Avoid pressing down on the patties with the spatula.
During the last minute of cooking, top each patty with a slice of cheddar cheese and cover the skillet to melt the cheese.
While the burgers are cooking, mix the mayonnaise, ketchup, and mustard in a small bowl to create the special sauce.
Lightly toast the brioche buns on the skillet or in a toaster.
Spread the special sauce on the inside of both the top and bottom halves of the toasted buns.
Assemble the burgers: place a cooked patty with melted cheese on the bottom bun. Top with shredded lettuce, tomato slices, and red onion slices.
Place the top bun on the assembled burger and serve immediately.