- For the gyro: Slice pork lengthwise into 4 equal steaks, about 1 inch thick (if steaks are thicker than 1 inch, press them with your hand to 1-inch thickness). Place pork in 1-gallon zipper-lock bag. Whisk ½ cup oil, garlic, oregano, salt, coriander, paprika, and pepper together in bowl. Add marinade to bag with pork. Seal bag and turn to distribute marinade evenly. Refrigerate for at least 1 hour or up to 24 hours.
- Adjust oven rack 6 inches from broiler element and heat oven to 350 degrees. Line rimmed baking sheet with aluminum foil and set wire rack in sheet. Remove pork from marinade and place on prepared wire rack; discard marinade.
- Cover sheet tightly with foil and transfer to oven. Roast until pork registers 100 degrees, about 40 minutes. Remove sheet from oven and carefully remove foil so steam escapes away from you. Return sheet to oven and continue to roast until pork registers 160 degrees, 20 to 25 minutes longer.
- For the tzatziki: While pork roasts, shred half of cucumber on large holes of box grater. Combine shredded cucumber, yogurt, oil, lemon juice, dill, salt, garlic, and pepper in bowl. Refrigerate until ready to serve. Slice remaining cucumber thin and set aside.
- Leave sheet in oven and turn on broiler. Broil until pork is well browned on top side only, about 10 minutes. Transfer pork to carving board and let rest for 5 minutes.
- Brush 1 side of pitas with remaining 1 tablespoon oil. Place 4 pitas oiled side up on now-empty wire rack. Broil until pitas are soft and lightly toasted, about 1 minute. Repeat with remaining 4 pitas.
- Slice pork crosswise very thin. Toss pork with accumulated juices on carving board. Divide pork evenly among pitas and top with lettuce, onion, sliced cucumber, and tzatziki. Serve with lemon wedges.
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