Begin by seasoning each lamb shank generously with salt and pepper on all sides.
Heat half of the olive oil in a large skillet over medium-high heat. Sear each lamb shank for approximately 5 minutes, ensuring a nice sear on all sides. Once seared, transfer them to a plate.
Prepare the garlic by smashing the cloves, discarding the skins, and slicing the garlic into slivers. Using a knife, create small slits in the lamb and insert the slivers of garlic into the meat.
In a mixing bowl, combine potatoes, cheese (such as Feta, Graviera, or Kasseri), onions, tomatoes, oregano, paprika, olive oil, any leftover garlic slivers, and lemon juice. Make sure to season the mixture generously with salt and pepper. Toss to coat the ingredients thoroughly and set the bowl aside.
Take two long pieces of parchment paper, each approximately 12 inches in length, and lay them in a crisscross pattern. Place a seared lamb shank in the center of each parchment paper assembly and arrange the seasoned vegetables around the lamb. Drizzle with additional olive oil.
Add two sprigs of thyme on top of each lamb shank and season further if desired. Repeat this process with the remaining lamb shanks to create a total of four parcels.
Fold the parchment paper flaps toward the center, then twist and secure the paper to create a sealed parcel. Use a piece of string or twine to secure the parcels. Place the secured parcels on a baking sheet for cooking.