The Spanish chorizo can be bulk or in casings. If bulk, just crumble it apart as fine as you can. If in casings, remove the casings and chop or crumble apart as fine as you can. One thought, if you grind your own beef, simply run the casing version, with the casings removed, through your grinder.
In a large mixing bowl, add the beef, chorizo, onion, and garlic. Mix well but do not over work the meat, over worked ground beef = tough burgers.
Divide the meat into 4, 5, or 6 patties, make a small indent in center of each patty with your thumb, this prevents the burger from going dome shaped or curling.
Heat a thick walled non stick pan and melt the butter. You want the pan hot completely.
Cook in batches if needed. Add patties to the pan and cook 5 minutes on the first side, flip and cook 4 more minutes for medium rare burgers. If you prefer burgers more done than that, just increase the times by a minute or so.
Remove to plates, top each burger with 1/2 tablespoon of mayo and then some of the cucumber slaw. Corn on the cob and potato salad come to mind for great sides.