Combine yogurt, olive oil, lemon juice, garlic and pepper in a bowl. Cover and refrigerate for one hour. Whip yogurt mixture until smooth. Dry cucumbers by patting with paper towels. Add cucumbers and dill to yogurt mixture. Salt and pepper to taste.
Mix lamb and beef in a glass bowl. Crumb bread into mixture. Add spices and mix well. Stir egg into mixture. Form mixture into two large patties and place on a medium-hot grill for 5 minutes per side or until the gyros is cooked through.
Serve sliced gyros on warm pita bread with fresh sliced tomatoes, onions and Tzatziki sauce.