Using your hands, combine ground venison or beef, ground bacon, ¾ teaspoon salt and ½ teaspoon black pepper in a large bowl until well combined. Form meat mixture into 4 (5-ounce) balls, then press into ¾-inch-thick patties about 4 inches in diameter.
If using canned chilies, skip to step 3. If you are using fresh peppers, put them on a rimmed baking sheet and place them under a preheated broiler in your oven until skins turn black, about 5 minutes. Place the charred peppers in a small mixing bowl and cover with plastic wrap for about 5 minutes. Using a paper towel, remove burnt skins and discard the skins. Slice tops off chiles, remove seeds, dice, and set aside.
In a small saucepan, melt 2 Tablespoons of the butter over medium heat. Add diced onion, garlic, and remaining ¼ teaspoon kosher salt. Sauté for about 4 minutes until onions are softened.
Stir in cumin and flour, mix well to combine, and cook another 2 minutes. Stir in diced chiles and another 1 Tablespoon of butter and cook another 1 minute.
Slowly add chicken broth while constantly stirring, reduce heat to low and simmer for 5 minutes, then turn off heat, but keep warm on the stovetop while you cook the burgers.
Brush cut sides of the hamburger buns with remaining butter and set aside.
Heat oil in a large (12-inch) skillet over medium-high heat until hot and shimmering but not smoking, then add the burger patties. Cook for about 2 minutes until well browned on first side. Flip patties, turn heat down to medium and cook another 2 minutes. Now reduce the heat to low and top each burger patty with 1 Tablespoon of the chile salsa mixture and 2 Tablespoons shredded Monterey Jack cheese. Cook until cheese is melted and patties reach an internal temperature reaches 145 degrees F using an instant-read meat thermometer, about 2 minutes. Transfer to large plate and let the burgers rest for 5 minutes.
Meanwhile, arrange buttered buns, cut side up, on a baking sheet. Broil in preheated oven until toasted, 1 to 2 minutes. Spread each bottom bun half with 1 Tablespoon mayonnaise, and top with a burger patty. Cover with top bun halves and serve immediately.