In a large bowl, combine the grated onion, garlic, yogurt, olive oil, tomato paste, paprika, cumin, salt, black pepper, thyme, and lemon juice.
Add the sliced meat and mix thoroughly until every piece is coated with the marinade.
Cover and refrigerate for at least 4 hours, preferably overnight.
Transfer the marinated meat onto plastic wrap and shape it into a compact log.
Wrap tightly and freeze for 1–2 hours until firm enough to slice.
Using a sharp knife, slice the chilled meat as thinly as possible.
Heat a cast iron skillet over medium-high heat.
Cook the meat in small batches without overcrowding the pan.
Turn frequently until browned with slightly crispy edges.
Warm the lavash or Turkish bread.
Fill the bread with döner meat, tomatoes, sumac onions, pickles, lettuce, and yogurt sauce if desired.
Serve immediately while hot.