In a large bowl, mix together the venison, bacon, egg, dijon, Worcestershire, onion powder, garlic powder, salt and pepper. Shape into 4 burger patties that are slightly larger than the buns. Let rest for about 10-20 minutes. This helps the burgers hold together when transferring to the stove or grill.
Stovetop: Heat a large cast iron skillet over medium-high heat with 1 tablespoon of butter. Once hot, fry the burgers about 3-5 minutes per side, about 6-7 minutes total for medium doneness. Work in batches if needed, so you don't crowd the pan. Add more butter as needed. Top the burgers with cheese during the last 1-2 minutes.
Grill: Preheat the grill over medium-high heat, and scrape the grate clean so the burgers don't stick. Once hot, grill each patty about 3-5 minutes per side, about 6-7 minutes total for medium doneness. Top the burgers with cheese during the last 1-2 minutes.
Toast the buns: If desired, spread butter on the insides of the top and bottom buns and place butter side down in the skillet or on the grill. Toast for 1-2 minutes until golden brown; watch closely so they don't burn.
Smoky Aioli: In a small bowl, mix together all ingredients.
Spread the bottom buns with smoky aioli. Add the burgers and your favorite toppings.