Grind partially frozen venison meat and pork butt together using a meat grinder or food processor; transfer to a large bowl and cover with plastic wrap. Refrigerate for at least 1 hour.
Heat unsalted butter in a skillet over medium-low heat; cook and stir onion, mushrooms, and 1 pinch salt in the melted butter until onion is soft and translucent, 5 to 10 minutes. Remove from heat and cool.
Gently mix ground venison-pork mixture, onion mixture, yogurt, 1 tablespoon Worcestershire sauce, and black pepper together in a bowl until just combined. Form into 6 patties and arrange on a large plate. Cover plate with plastic wrap and refrigerate for at least 1 hour.
Mix mayonnaise, steak sauce, 1 tablespoon Worcestershire sauce, and lime juice together in a bowl until smooth. Cover bowl with plastic wrap and refrigerate until serving time.
Preheat oven to 225 degrees F (110 degrees C).
Spread 1/2 teaspoon butter onto the inside of each split bun.
Heat a non-stick pan over low heat; cook buns, buttered sides down, in the heated pan until lightly browned, 3 to 5 minutes. Transfer buns to a baking sheet and keep warm in the preheated oven.
Heat a non-stick skillet over medium heat; cook burgers, sprinkling each side with a pinch of salt, in the hot skillet until your desired degree of doneness is reached, 3 to 4 minutes per side. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C).
Top each burger with a tomato slice and two slices pepper Jack cheese. Remove skillet from heat and cover pan until cheese is slightly melted, about 5 minutes. Spread 1 to 2 tablespoons of sauce onto the inside of each bun bottom. Add a cheese-tomato burger to each bun bottom; top each burger with bun top.