To make the slaw: mix all the slaw ingredients together. Set aside for at least 30 minutes to let flavors meld.
Prepare charcoal grill for medium-hot direct-heat cooking.
To make the Chili Rojo: add all chili ingredients, except extra wine, to a blender or food processor. Puree thoroughly, using extra wine if necessary to make a thick, smooth sauce.
To make the burger patties: take 6 tablespoons ground beef and mix with 6 tablespoons Chile Rojo. Make 6 equal hand-formed patties with the remaining ground beef. Place 2 tablespoons of the chile mixture onto the center of each patty. Carefully enclose each patty around the chile mixture to form a ball. Carefully flatten each ball back into a thick patty, keeping the chile mixture inside. Season the patties liberally with salt and pepper.
To make the Crema: mash the avocados in a food processor or with a potato masher. Add the rest of the ingredients and fold gently until thoroughly mixed.
When the grill is hot, oil the grill rack and cook the burgers over a medium-high flame until done, about 3 minutes per side for medium rare. If the exterior of the patties begins to burn, move them to a cooler side of the grill until the centers have had a chance to cook. Add a slice of cheese to each patty about 1 minute before they are done. Let patties rest off the heat for 1 or 2 minutes after cooking, and meanwhile toast the buns on a cooler side of the grill.
To assemble the burgers: spread each bun bottom with a layer of Avocado Crema. Add about 1/3 cup slaw, a patty with melted cheese, then a tablespoon of chili ketchup, and finally the bun top.
I like the "whole cow" approach to ground beef and use 2 parts chuck to 1 part rib meat to 1 part sirloin or round. This gives the mixture a variety of texture and beefy flavor. If you are limited in choices, chuck is by far the best meat for grinding and for burgers.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.