Make the Buns:
Add warm water, milk, egg, softened butter, sugar, salt, bread flour, and yeast to the bread machine pan in that order. Select the dough cycle and run until complete (about 1.5 hours).
Turn dough onto a floured surface. Divide into 8 equal portions (100–110 g each). Shape into smooth balls, then flatten into 3.5–4 inch (9–10 cm) discs. Place on a parchment-lined baking sheet.
Cover and let rise in a warm spot for 45–60 minutes, until puffed.
Preheat oven to 375°F (190°C). Brush buns with beaten egg or milk, sprinkle with sesame seeds if desired.
Bake for 15–18 minutes until golden brown. Cool on a wire rack.
Make the Patties:
In a large bowl, combine ground venison, pork fat or grated butter, egg, breadcrumbs, Worcestershire sauce, garlic powder, onion powder, paprika, pepper, and balsamic vinegar. Mix gently until just combined — do not overmix.
Divide mixture into 6–8 portions (4–5 oz or 113–142 g each). Shape into patties about 3/4 inch (2 cm) thick, slightly wider than the buns. Press a shallow thumbprint into the center of each.
Chill patties on a tray for at least 30 minutes (up to 2 hours).
Season both sides with salt right before cooking.
Cook:
Preheat a grill to medium-high (375–400°F / 190–204°C) or heat a lightly oiled cast-iron skillet over medium heat.
Cook patties 4–5 minutes per side for medium doneness (internal temp 160°F / 71°C). Add cheese in the last minute if using. Avoid pressing the patties.
Rest cooked patties 3–5 minutes under loose foil.
Toast buns lightly, then assemble burgers with toppings of choice. Serve warm.