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Prepare the Tzatziki Sauce:
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In a medium bowl, combine the Greek yogurt, grated cucumber, minced garlic, lemon juice, dill, salt, and pepper. Mix well and refrigerate until ready to use.
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Cook the Ground Beef:
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In a large skillet, heat 1 tablespoon of olive oil over medium-high heat.
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Add the ground beef and season with garlic powder, onion powder, dried oregano, ground cumin, paprika, salt, and pepper.
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Cook the beef, breaking it up with a spoon, until it is browned and cooked through. Remove from the skillet and set aside.
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Sauté the Peppers and Onions:
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In the same skillet, heat another tablespoon of olive oil over medium-high heat.
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Add the sliced onions and bell peppers, season with salt and pepper, and sauté until they are tender and slightly caramelized, about 5-7 minutes.
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Assemble the Gyros:
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Warm the pita breads in a dry skillet or in the oven.
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Spread a generous amount of tzatziki sauce on each pita bread.
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Add a portion of the cooked ground beef, followed by the sautéed peppers and onions.
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Top with halved cherry tomatoes, crumbled feta cheese, and fresh lettuce leaves, if desired.
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Serve:
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Fold the pita bread around the filling and serve immediately.