Preheat oven to 475°F with a rack about 6 inches from heating element. Place skewers in a shallow dish; cover with water, and let stand until ready to use.

Stir together ground beef, onion, tomato, shawarma spice blend, ½ teaspoon grated garlic, 1 tablespoon parsley, 1 ½ teaspoons salt, and ½ teaspoon pepper in a large bowl until mixture is very well combined and slightly sticky, about 1 minute.

Lay tortillas on a clean work surface. Divide beef mixture evenly (about ¾ cup each) among tortillas, spreading in a very thin, even layer all the way to edges. If desired, place a piece of parchment paper directly on meat, and use your hands or a rolling pin to help spread meat over tortillas.

Roll tortillas tightly into logs, pressing gently as you roll to compact filling. Chill 20 minutes for cleaner slicing, if desired. Slice each log into 1-inch-thick pinwheels. Discard thinner ends. Thread 4 to 5 pinwheels onto each skewer, cut sides facing out and pieces touching, so they hold their shape.

Spray a baking sheet with cooking spray. Place skewers, cut-side down, on prepared baking sheet and bake until starting to brown, about 6 minutes. Flip, and continue to bake until starting to crisp in spots, about 4 minutes. Increase oven temperature to broil, and continue cooking until edges are browned and crispy in spots, 2 to 4 minutes, turning once halfway through.

While pinwheels cook, stir together yogurt, lemon juice, lemon zest, dill, remaining 1 tablespoon parsley, ½ teaspoon grated garlic, ½ teaspoon salt, and ¼ teaspoon pepper in a small bowl until smooth.

Transfer pinwheels to a platter. Garnish with feta, parsley, and dill. Serve with lemon wedges and yogurt sauce.