Homemade Doner Kebab

Name: Homemade Doner Kebab
Ingredients: For the Kebab:; 1 pound ground lamb (or 1/2 pound each of ground lamb and ground beef); 1 egg; 4 cloves garlic (peeled and finely minced); 1 teaspoon ground cumin; 1 teaspoon ground coriander; 1 teaspoon smoked paprika; 1 teaspoon dried oregano; 1/2 teaspoon salt; 1/4 teaspoon ground black pepper; 1 tablespoon oil (for the skillet, plus extra for the pan); For the Sandwich:; 4 rounds of pita (or naan or flatbread); 1 cup assorted lettuce; 1 large tomato (sliced); 1/2 seedless English cucumber (sliced); 1/4 large red onion (peeled and sliced); Tzatziki sauce or tahini sauce