To make the whipped feta, crumble the feta into the work bowl of a food processor. Add the yogurt and olive oil and pulse until light and creamy, stopping to scrape down the sides a few times to process evenly.
To make the pickled fennel, place the sliced fennel and red onion in a small stainless steel bowl. Combine the vinegar, water, sugar, salt and seasoning in small saucepan. Bring to a boil and pour over the fennel. Allow to cool before serving or refrigerate.
Heat a grill over high heat. Combine ground lamb, oregano, garlic and salt and pepper. Mix by hand to evenly distribute the ingredients. Portion into 6 balls. Flatten into patties slightly larger than the bun. Grill over high heat for 4-5 minutes per side for medium or to desired doneness. Remove from the grill. Cut the buns in half, spread the feta on the bottom bun, top with a little arugula, a slice or two of tomato, some of the pickled fennel and onions, the lamb patty and a small spoonful of the feta. Place the top bun on top and secure with a toothpick.
Serve with a glass of Cakebread Cellars Rubaiyat.