In a large bowl, combine the garlic, lemon juice, red wine vinegar, olive oil, yogurt, oregano, salt and pepper in a medium bowl and whisk well. Add the beef pieces to the bowl and mix well to coat. Cover bowl or transfer into ziploc bag and refrigerate for about 1 hour.
To make the tzatziki sauce:
While the beef is marinating, shred the cucumber or chop in food processor. Toss with 1/2 teaspoon salt and set over a strainer or paper towels (a coffee filter also works well) and let sit for 20 to 30 minutes. Squeeze cucumber to remove as much liquid as possible.
In a medium bowl, combine yogurt, cucumber, lemon juice, dill, garlic, and white wine vinegar. Season to taste with salt and pepper. Drizzle lightly with olive oil. Refrigerate for at least 30 minutes.
To cook the beef in the oven:
Preheat oven to 400 degrees. Coat a baking dish with oil or nonstick spray and arrange beef in a single layer. Cover with parchment paper, tucking the paper around the beef so it is completely covered.
Bake for 12 to 18 minutes, until the beef reaches 140 degrees when tested with an internal thermometer. Remove from oven and let rest for 5 minutes. Slice or chop into pieces.
Serve on pita bread with fresh dill, feta cheese, tomatoes, red onion, tzatziki sauce, and fresh dill.
To cook the beef on the grill:
Preheat grill on medium-high heat. Grill the beef turning occasionally until the internal temperature reaches 140 degrees on a thermometer, about 8 to 10 minutes.
Remove from grill and let rest for 5 minutes. Slice or chop into pieces. Serve on pita bread with feta cheese, tomatoes, red onion, and tzatziki sauce.
Plain Greek yogurt: Any fat percentage will do, but I adore the richness that you can only find in a 2 to 5 percent variety. (Translation: Use the 0 percent kind with caution.)
Beef flank steak: While traditional gyros are made with lamb, beef is a wonderful, crowd-pleasing alternative. Flank steak is widely available and budget-friendly. If desired, swap in skirt steak, hanger steak, or flat iron steak.
Cucumber: Peel, grate, and strain this before incorporating into your tzatziki sauce so the condiment doesn't get too watery.
Yield: This recipe makes enough to build 4 generous gyros with approximately 4 ounces of beef per gyro.
Storage: Store leftovers covered in the refrigerator for up to 4 days.