In a small bowl, stir all of the sauce ingredients until smooth, then set aside.
For the Venison Burgers
Preheat a grill to medium high.
In a large bowl, combine ground venison, salt, pepper, and 3 Tablespoons of the melted butter. Using clean hands, stir until well combined.
Divide the ground venison into four 4-ounce portions. Gather each into a ball, then gently flatten into a 4-inch-wide patty about ¾-inch thick. Press your thumb into the center of each patty to make a ¼-inch-deep indentation (this prevents the patties from bulging as they cook).
Lightly brush the grill grates with vegetable oil, then grill the patties indentation-side up until marked on the bottom, about 4 minutes. Flip the burgers only once and cook until marked and slightly firm, about 2 more minutes until instant-read meat thermometer reaches 130 degrees F. Check the temperature often so you don't overcook and dry them out!
While the burgers are cooking on the grill, open the hamburger buns and brush both sides liberally with the remaining 3 Tablespoons of melted butter.
Remove the venison burgers to a plate, top with a slice of American cheese, and let burgers rest for 5 minutes to let the juices redistribute.
Meanwhile, place the buns onto the hot grill, buttered-sides down, and cook until golden brown, about 2 minutes. Transfer buns to a plate and spoon 1 tablespoon of sauce on each bottom bun, along with a piece of lettuce and two slices of tomato. Spoon another tablespoon of sauce on each top bun.
Transfer a venison cheeseburger to each bottom bun, then add the top bun with red onion and pickles. Serve immediately.