Add sliced onions to a saute pan set over medium heat along with one tablespoon of butter. Slowly caramelize the onions until they turn brown and sticky, stirring on occasion to prevent burning. *
When onions have finished caramelizing stir the red wine vinegar until it is absorbed by the onions. Then remove the caramelized onions from the skillet and set them aside.
Form your venison burger patties while the onions are caramelizing. Do this by separating your ground venison into ⅓ pound chunks. Then press the meat flat with the bottom of a plate until it forms a burger patty slightly larger than the width of your buns. Season each patty with salt and pepper to taste.
Add one tablespoon of butter into a large skillet set over medium-high heat and add the raw venison patties. Cook until the patty is well seared underneath, then flip once to cook the other side and place the cheese on top to melt as it finishes cooking. Do not press the patties as they cook! Cook to an internal temperature of 160 degrees.
While the burgers are cooking, slice your hamburger buns in half and brush the cut interior with butter then lightly toast the buns in a separate skillet until they begin to turn golden brown.
Prepare the burger sauce by mixing the mayo, steak sauce, and Dijon mustard together in a small bowl until well combined.
Assemble your juicy burgers by placing lettuce on the bottom bun, followed by the burger patty, cheese, caramelized onions, burger sauce, and the top bun. Serve immediately.