Pork Souvlaki with Tzatziki

trimmed pork shoulder, cut into 3-by- 1/2-inch strips

large onion, cut through the root end into 1/2-inch wedges

plus 2 tablespoons extra-virgin olive oil

fresh lemon juice, plus lemon wedges for serving

chopped fresh oregano

garlic cloves, mashed to a paste

Kosher salt and freshly ground pepper

Greek-style whole-milk yogurt

European cucumber, seeded and finely diced

chopped fresh mint

Warm pita, for serving