Spray a baking sheet with nonstick cooking spray. In a medium mixing bowl, whisk the grated onion, garlic, egg, salt and pepper. Add the ground beef and use your hands to gently mix until combined. Divide the mixture evenly into 4 parts, then divide each into 2, to give you 8 total. Form each into a thin patty and place onto the prepared baking sheet. The mixture will be wet and a bit sticky. Refrigerate for 20 minutes.
Heat a griddle or large cast-iron skillet over medium-high heat. Add the butter and sliced onions and cook, stirring every so often, until the onions have a nice sear to them, 4 to 5 minutes. Set aside.
Wipe the griddle clean and return to medium-high heat. Put the patties on the griddle, working in batches if necessary. Spoon about 1 teaspoon mustard (or squeeze directly from the bottle) onto the top raw side of each patty. Cook about 3 minutes, then flip and cook an additional 3 minutes. Top each with a slice of American cheese.
Spread some Special Sauce on each bottom bun, then add lettuce, 1 patty, some onions, a second patty, tomatoes, pickles, more sauce and a top bun.
Combine the mayonnaise, mustard, relish, ketchup, vinegar, garlic powder, onion powder and paprika in a medium bowl and whisk until well blended.