Preheat oven to 325 degrees F (165 degrees C). Bring a large pot of salted water to a boil. Add potatoes and cook until tender but still firm, about 15 minutes. Drain, cool and chop.

Lightly brush a 9x13 inch pan with melted butter. Place 1 phyllo sheet in pan and lightly brush with butter. Continue adding phyllo sheets for a total of 12, brushing each one lightly with butter (you don't need to completely cover each phyllo sheet with butter).

Spread cooked rice over phyllo, then sprinkle with minced garlic. Add lamb in an even layer, and sprinkle with lemon juice. Place 1 cup diced potatoes over lamb.

Arrange quarters of eggs, then sprinkle with parsley, mint and lemon zest. Add crumbled feta cheese.

Pour on olive oil and beef broth. Sprinkle with oregano and pepper. Finally, add beaten egg.

Top with remaining 12 phyllo sheets brushed lightly with melted butter.

Bake for 40 to 50 minutes at 325 degrees F (165 degrees C). during the last 10 minutes, raise the temperature to 350 degrees F (175 degrees C). Remove from the oven and cool on a rack for 15 minutes. Cut into diamonds or squares and serve warm.