Brown the ground beef and chopped onion in a large skillet over medium heat. Drain off any excess grease.
Stir in the water, ketchup, yellow mustard, 1 tablespoon sweet pickle relish, 1 teaspoon white vinegar, onion powder, garlic powder, salt, and pepper into the skillet.
Bring the mixture to a simmer, then reduce heat to low. Cook for 10 minutes, stirring occasionally, until the sauce thickens slightly.
While the meat simmers, prepare the special sauce: In a small bowl, whisk together the mayonnaise, 2 tablespoons sweet pickle relish, minced dill pickle, 1 teaspoon white vinegar, and sugar.
Toast the hamburger buns lightly, if desired.
Spoon the meat mixture onto the bottom halves of the buns.
Top the meat with a slice of cheese, shredded lettuce, and sliced dill pickles.
Drizzle a generous amount of the homemade special sauce over the pickles.
Place the top bun on the sandwich. Serve immediately.
Notes
For a one-pan meal, skip toasting the buns separately; simply place the bottom buns in the skillet after the meat is done, top with cheese, cover briefly until cheese melts, then assemble.
If you prefer a tangier sauce, increase the white vinegar slightly in the special sauce mixture.
Use lean ground beef to reduce the amount of grease you need to drain.