First whip up the yogurt sauce by combining the yogurt, vinegar, lemon juice, garlic powder, and a pinch of kosher salt in the a small mixing bowl. Whisk to combine. Add the olive and whisk again until smooth. Chill while you make the rest of the quesadilla.
Next up, make a quick salad by combining the cucumber, red onion, and cherry tomatoes in a medium bowl. Toss with a heavy pinch of kosher salt, olive oil, and red wine vinegar and set aside.
To make the quesadilla, heat a large skillet or griddle over medium high heat. Add a splash of oil and the white onion. Sprinkle with a pinch of salt. Cook the onions until tender and just beginning to brown, about 5 minutes. Add the gyro meat and use a metal spatula to break up the meat into bite size pieces. Cook until the meat is hot and just crisping on the edges. Add the tortillas to the griddle and add the meat and a sprinkle of feta cheese. Add a second tortilla and flip carefully. Cook the quesadilla until browned and the cheese is just softened.
Off the heat, slice the quesadilla and top with the yogurt sauce and chopped salad. Garnish with dill, if desired, and serve.