Using an immersion circulator, preheat a water bath to 132° – 137°F.
Gently form four 6-ounce patties, being careful not to overwork the meat. Season both sides of each patty with salt and pepper.
Seal patties individually in vacuum bags. Cook patties in the water bath for 2-2.5 hours.
While the patties are cooking, make the burger sauce: in a small bowl, stir together mayonnaise, ketchup, sweet pickle relish, and vinegar. Cover and refrigerate until ready to use.
To finish, remove the patties from the bags and place on paper towels; carefully and thoroughly pat dry. Rest for about 10 minutes. Season with salt and pepper. Using a griddle, grill, or broiler, toast buns with butter, if desired.
Heat a lightly-oiled grill or cast-iron pan over a very high flame. Sear or grill until a nice crust has formed, about 1 minute. Flip the patties over and top with cheese. Continue to sear about 1 minute more.
Dress buns with reserved burger sauce, lettuce, tomato, and/or pickle (if using), patties, and bun tops. Serve immediately.