In a large mixing bowl, combine ground beef, crumbled feta cheese, beaten egg, chopped onion, minced garlic, oregano, salt, and pepper. Mix gently until incorporated.

Divide the beef mixture into four portions and shape each into thick patties, about 1 inch thick.

Heat olive oil in a skillet over medium-high heat. Cook the patties for 4-5 minutes on each side until golden-brown.

Sauté minced garlic in the skillet for about 30 seconds until fragrant.

Stir in tomato paste and cook for 1 minute. Whisk in heavy cream, add crumbled feta and smoked paprika. Cook until thickened.

Return the patties to the skillet, spoon sauce over them, and heat together for 1-2 minutes.

Plate the burgers, drizzle with additional sauce, and garnish with chopped parsley and crumbled feta.