Puree the onion in a food processor until smooth. Strain through a fine-mesh sieve over a medium bowl. Whisk the oregano, cumin and 1/2 teaspoon each kosher salt and pepper into the onion juices. Add the lamb; toss. Cover and refrigerate at least 1 hour or overnight.
Preheat a grill to medium high and oil the grates. Remove the lamb from the marinade and thread onto 4 metal skewers. Grill, turning occasionally, about 8 minutes for medium doneness. Sprinkle with flaky salt and serve with tzatziki.