Marinate pork: Whisk together 3/4 cup oil, 2 tsp. each salt and dried oregano, the lemon zest and garlic, 2 tbsp. lemon juice, the onion powder, and parsley in a wide medium bowl. "Lemon, dried oregano, and olive oil--that's the holy trinity of Greek cooking," Bililies says. "Dried oregano is spicier than fresh." Add red onion and pork and turn to coat thoroughly. Marinate at least 1 hour (chill if longer than 1 hour) and up to 24 hours.
Grill souvlaki: If using a charcoal grill, light a chimney of briquets. When just covered with ash, scatter evenly over firegrate. If using gas, heat grill to medium-high (450°).
While grill heats, whisk together remaining 3 tbsp. oil, 1 tbsp. lemon juice, and 1/2 tsp. each salt and dried oregano to make a vinaigrette. Set aside. Thread 5 or 6 pork cubes onto each skewer, adding an onion piece between them, and adding a bay leaf, if using, after every 2 pork cubes. "I'll put some of the fattier pieces on the end for an extra-delicious first bite," says Cervone. Discard marinade. Cut 6 pieces of parchment or waxed paper (each about 12 in. square) for wrapping sandwiches and set aside.
Grill pita, following step 6 of recipe, and cover with a kitchen towel to keep warm.
Grill skewers (covered on gas), turning to brown all sides, until done the way you like (for medium*, that means about 8 minutes total on charcoal, and 10 minutes if using gas). Move hot souvlaki to a platter and drizzle with lemon-oregano vinaigrette "to brighten it up and round out the flavor," says Cervone. Warm up pita if they've gotten cold, advises Bililies. "They're more flexible when they're warm."
Assemble sandwiches: Snap bay leaves off skewers, if using. Arrange a sheet of parchment on a work surface so it looks like a diamond. Set a warm pita on top corner of diamond so it extends a little beyond parchment. Spread with 2 to 3 tbsp. yogurt sauce--"You don't need to go super crazy with it," says Bililies--then push off a skewer's worth of pork down its center. Top pork with 1/2 cup cucumber-tomato salad, then 1/4 cup pickled onions. Finish with 2 tbsp. feta.
Using parchment to help guide, fold sides of filled pita over each other. Wrap parchment around sides of sandwich and fold end of paper under sandwich. Fill and wrap rest of pita the same way.
Set out hot sauce for drizzling. "I can see all the Greeks having a fit right now," says Bililies. "They don't do spicy. But Americans love it."
*Pork in the United States no longer has trichinosis, so it doesn't need to be cooked until well done and dry, as in the past.
Make ahead: Pork in marinade, up to 1 day, chilled. For other make-aheads, see individual recipes.
Note: Nutritional analysis is per serving (pork only).