Combine yogurt, olive oil, lemon juice, garlic, and white pepper in a bowl for the tzatziki sauce.
Cover and refrigerate the yogurt mixture for one hour.
Blend the chilled yogurt mixture with a whip until smooth.
Dry the diced cucumbers by patting with paper towels.
Add cucumbers and dill to the yogurt mixture, then season with salt and pepper to taste.
Mix lamb and beef together in a glass bowl.
Crumb the day-old bread into the meat mixture.
Add all spices (garlic, parsley, salt, pepper, fenugreek, and cumin) and mix well.
Stir the beaten egg into the meat mixture until well combined.
Form the mixture into 2 large patties.
Place patties on a medium-hot grill and cook for 5 minutes per side or until cooked through.
Slice the cooked gyros meat and serve on warm pita bread with fresh sliced tomatoes, onions, and tzatziki sauce.