Combine all of the mint sauce ingredients in a food processor or blender and blend until combined. Refrigerate until ready to use.
In a bowl, combine the lamb, dried oregano, and cumin with salt and pepper to taste. Using your hands, knead the meat until the spices are just combined (don't overwork the meat).
Heat a cast-iron skillet or grill over medium-high heat. If I am using a cast iron, I like to spray avocado oil or olive oil on the bottom of the skillet. Grill the burgers on both sides until it is cooked to your liking, I did mine for about 4 minutes per side in a cast-iron skillet.
Serve burgers over baby arugula, sliced tomatoes, some marinated cucumbers and red onions (optional), and with the mint sauce.