First, process the onion in a food processor, high power blender or 10 to 15 seconds or grate the onion.
Place the grated onion into a paper towel-lined colander and press until almost all of the onion juice is removed. Discard juice.
Place all of the gyro meat ingredients into a mixer fixed with the paddle attachment.
Blend the ingredients on medium speed for 4 minutes. The meat will be well blended and almost gummy.
Form the meat mixture into a cylinder shape and place on a double thickness of plastic wrap that are at least 18 inches long.
Roll the mixture in the plastic wrap tightly, making sure to remove any air pockets. Once the meat is completely rolled in the wrap, twist the ends of the plastic wrap until the surface of the wrap is tight.
Store in the refrigerator for at least 2 hours or up to overnight, to allow the mixture to firm up.
Preheat the grill to high.
Once fully chilled, slide the meat onto the rotisserie skewer and secure it.
Place a double-thick piece of aluminum foil folded into a tray directly under the meat to catch any drippings.
Cook on high for 15 minutes. Decrease the heat to medium and continue to cook for another 20 to 30 minutes or until the internal temperature of the meat reaches 165 degrees F.
Turn off the heat and allow to continue to spin for another 10 to 15 minutes or until the internal temperature reaches 175 degrees.
Slice and serve on pita bread with tzatziki sauce, chopped onion, tomatoes, and feta cheese.