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Preheat oven to 375°F (190°C). Line a baking sheet with parchment paper or lightly grease a baking dish.
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In a large bowl, combine the ground beef, egg, milk, breadcrumbs, onion, garlic, salt, pepper, paprika, and oregano.
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Add the shredded cheese and ketchup. Mix gently just until combined—do not overmix.
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Shape the mixture into flat, round patties and place them on the prepared baking sheet.
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Brush the tops lightly with olive oil.
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Bake uncovered for 20 minutes.
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Remove from the oven, sprinkle extra cheese on top if desired, and return to the oven for 5 more minutes, or until the cheese is melted and patties are fully cooked.
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Let rest for 5 minutes before serving.
