FINELY mince one medium onion by hand, or by pulsing in a food processor; PLACE onions into a heavy-bottomed non-stick or stainless steel skillet; ADD 2 tablespoons water and 3/8 teaspoon kosher salt; COOK onions over medium heat until onions are translucent and most water has evaporated (replenish water if necessary to keep onions just moistened); SPREAD onions evenly and levelly in a 15 x 12 baking sheet; LOOSELY cover; PLACE sheet of onions into freezer until almost frozen, adhering to bottom of baking sheet.
START with a clean oven; PLACE oven rack in the lower 1/3 position; PREHEAT oven to 500°F ; In a medium bowl, mix 3/4 lb ground beef with 1 1/2 tablespoons water, 3/8 teaspoon salt and 1/8 teaspoon ground black pepper in a medium bowl with a fork until thoroughly combined; DUMP ground beef mixture onto a COVER ground beef mixture with a sheet of waxed paper; PRESS ground beef mixture into the baking sheet with onions evenly to the edges using hands first, then with a rolling pin if desired, for uniformity; REMOVE waxed paper and discard.
CUT ground beef lightly into 12 symmetrical squares using a pizza cutter (3 across x 4 down) leaving meat intact; POKE 5 evenly-spaced holes into each beef patty square (resembling the 5-dot pattern of standard playing dice), with a thin wooden spoon end or drinking straw, pressing while twisting to bore holes.
PLACE squeezable ketchup and/or mustard, sliced pickles and sliced cheese within easy reach in work station to assemble Slyders quickly when needed; TRIM or cut American cheese to fit Slyders (I usually trim the edges only, but to economize you may cut cheese slices into 4 small portions).
DIVIDE one package (12) Sara Lee Classic Dinner Rolls using a sharp knife for crisp clean edges; CUT each roll to a 1/3 slice for the bottom half, leaving 2/3 for the top bun; REPLACE bun tops; ARRANGE buns on a clean platter.
INSERT a second baking sheet, inverted, over prepared baking sheet with meat and onions to prevent spatter and to help steam; CAREFULLY transfer the baking sheets containing beef mixture into preheated oven wearing oven mitts; CLOSE oven door.
STEAM-BAKE beef patties for about 7-8 minutes or until beef is no longer pink (patties will shrink slightly upon cooking); REMOVE from oven; MOVE baking sheet with cooked patties into work station.
LIFT each beef patty with a slotted spatula and place over bottom buns; PLACE a slice of cheese over each onion-topped meat patty diagonally; ARRANGE dill pickle slices over cheese.
SQUEEZE 1 teaspoon mustard OR 1 teaspoon ketchup evenly onto inner TOP bun; PLACE one meat patty over each BOTTOM bun using a slotted spatula; SET bun tops atop assembled Slyders.
WARM prepared Slyers on platter in microwave at 100% power only until warmed and cheese has melted (about 15 seconds); SERVE warm Slyders immediately and enjoy!