Place feta and cream cheese in a food processor and pulse until the cheeses are mixed. Add garlic, lemon zest, lemon juice, and heavy cream, and process until completely smooth, adding more heavy cream if necessary to achieve desired consistency. Keep refrigerated until ready for serving.
Meanwhile, combine ground beef, shallots, garlic, parsley, oregano, Worcestershire sauce, and black pepper in a large bowl until well mixed. Form into 4 patties.
Grill or pan fry the patties over medium-high heat for approximately 3 minutes on each side or until they are cooked to desired level of doneness.
Serve on burger buns with toppings and whipped feta spread. Enjoy!