In a large mixing bowl, combine ground beef, crumbled feta, egg, chopped onion, minced garlic, dried oregano, salt, and pepper. Gently blend until just combined.

Divide the mixture into four equal portions and shape each into a patty about 1 inch thick.

Heat a tablespoon of olive oil in a skillet over medium-high heat. Cook the patties for 4-5 minutes on one side until browned. Flip and continue cooking for another 4-5 minutes until cooked through.

In the same skillet, add more olive oil if needed and sauté minced garlic for about 30 seconds until fragrant.

Stir in tomato paste and cook for 1 minute, then gradually whisk in heavy cream until smooth and thickened.

Mix in smoked paprika and any remaining feta. Return the patties to the skillet, and spoon sauce over them. Cook for an additional 1-2 minutes.

Plate the burgers with the sauce, optionally garnished with fresh parsley and feta crumbles, and serve.