For the burgers: In a bowl, blend the beef and lamb. Sprinkle with salt and pepper, and then remove from the bowl and form 4 patties, 8 ounces each.
Next, to cook the burgers, warm a large pan with the oil over high heat until the verge of smoking. Then reduce the heat to medium and add the burgers. Cook on the first side for 5 minutes, and then flip and repeat the process on the second side. During the second side of cooking, top with the feta cheese and finish for final time.
For the sauce: During the burger cooking, in a bowl blend and whisk together the sour cream, cucumbers, onions, tomatoes, salt and pepper. Taste and add any additional seasoning if desired.
Warm a second pan over medium heat. Spread the butter on the interior of the buns, and then place in the warmed pan, browning until crusted, 2 to 3 minutes.
Once cooked, remove and build the burgers with lettuce, tomato, burger, and top with sauce and enjoy.