Sauté the chopped shallots in the butter until soft. Set aside to cool.
Add all ingredients to the meat in a large mixing bowl. Mix well.
Divide ground venison into 4 even size pieces and, using wet hands, shape into burger patties to match the bun size (about ¾ inches thick).
Keep cold until ready to cook. Garnish.
Use butter to grease a large, hot, heavy pad or flat-top grill.
Grill the burgers and the onion until well colored on each side and continue until your desired degree of cooking. Medium rare is about 3 minutes on each side.
Place cheese and onions on hot burger and leave to let meat rest. Fry egg until just soft.
Toast the buns on the cut side. Spread sauce on the bottom bun. Place the burger, onions, then the egg on top. Add the sliced beet and then lettuce. Finish with top bun and press lightly to secure and serve.