In a large bowl combine the ground beef, chorizo, green chilies and grated cheese. Mix thoroughly. Form into four fat patties with the same diameter as the hamburger buns (they will be almost an inch thick/high). Place on a plate or baking sheet, cover with plastic wrap and place in the refrigerator for at least an hour to chill.

Heat a gas grill to High. Once fully heated brush the grill clean with a grill brush tool. Divide the hamburger buns to tops and bottoms; place the inside of each half face down on the grill to mark the buns and warm them (30 seconds to a minute; check frequently so they do not burn). Set aside.

Lightly brush both sides of the onion slices with olive oil. Place full slices onto the grill and close the lid. Grill slices well on the first side then turn over, being careful to keep the slices intact as much as possible. Grill onions until they are translucent with grill markings (about 10 minutes). Remove from grill and set aside.

Brush grill once more. Lightly brush each side of the chilled hamburger patties with olive oil. Place them onto the grill allowing space between them. Close the lid of the grill and allow them to cook for 3 minutes and develop a good sear.

With a grill spatula, gently turn the burgers over. Close lid and allow the burgers to cook an additional 6-8 minutes, checking for doneness by the burger color (should be browned and opaque), any juices coming from the burger naturally being clear (note: the chorizo can lend an orange tint to the juice) and lastly checking for an internal temperature of 160 degrees. Remove burgers and place them on a clean plate for 5 minutes to rest.