Preheat BBQ grill to 375° F, (medium-high).
Prepare and measure ingredients to be combined with ground beef and ground lamb.
In a large bowl, add the ground beef and ground lamb. Pour evenly over the beef and lamb the chopped onion, minced garlic, olive oil, herbs and spices. Use your hands to mix the ingredients until they are combined well.
Place the hamburgers on the grill. Cook 4-5 minutes on the first side. Flip the hamburgers over and cook an additional 4-5 minutes. Use a meat thermometer to check the temperature. The burgers will be medium-well when they reach 160° F.
Brush buns with extra virgin olive oil and place on grill for 2-3 minutes to brown.
Assemble burgers in this order: Bottom of bun, tzatziki, patty, tomato, arugula, onion, tzatziki, top of bun.