Cut meat into approximate 2″ cubes. Using the food grinder on your Kitchen Aid Stand Mixer, grind your burgers on the larger “coarse” dial into a medium bowl. Sometimes the grinder likes to spit out a bit of meat juice, so it helps to put a cup close to the dial to catch any spurts.
Using your hand(s), gently combine the meats, taking care not to compact the meat to much. Form burgers into 4 equal-sized patties and season liberally with Kosher salt.
If using a grill: Grill the burgers over high heat until medium rare, 3 to 5 minutes per side. Add a slice of cheese to each burger during the last 1-2 minutes of cooking. Brush buns with butter and grill until lightly toasted.
If using a cast iron skillet: preheat over high heat until very, very hot. Place butter in pan and cook burgers over high heat until medium rare, about 3-5 minutes per side. Add slice of cheese to each burger during the last 1-2 minutes of cooking. Remove burgers and add buns to toast slightly.
Let bugers rest for a few minutes, then place onto each bun and top with a few spoonfuls of coleslaw.
For the Coleslaw
In a small mixing bowl, whisk together the vinegar, mustard, mayonnaise, Worcestershire sauce, sugar, and salt until fully combine.
In a medium mixing bowl, mix together the cabbage, garlic, onion, and jalapeno. Pour the dressing over the slaw and toss gently to combine. Cover and refrigerate 1 hour.