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Place the ground lamb in a medium bowl. Add the olive oil, grated tomato, red wine, oregano, sea salt and pepper. Add the garlic, mint, parsley, mint, salt and pepper. Knead to combine. Divide into 4 portions and shape into burgers. Refrigerate for 15 to 30 minutes, to firm up.
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Whisk together the crumbled feta, yogurt, mint, vinegar, 1 minced clove of garlic, and salt and pepper to taste in a small bowl. Refrigerate sauce until ready to serve.
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Light the grill to medium and grill the burgers, turning just once, until cooked to desired doneness, about 4 to 5 minutes per side for medium to medium-well burgers.
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Lightly oil the insides of the burger buns (split them open!). Grill the buns open side down until lightly brown and marked with grill marks, for a minute or so per side. Top each bottom half of each bun with a little arugula, then with the burger. Spread a little of the yogurt - feta sauce over the burger and garnish with onion, tomato and cucumber or pickle slices. Serve.