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

Divide into four equal portions and shape into thick patties, making an indentation in the center of each.

Heat olive oil in a skillet over medium-high heat, and add the patties. Cook for 4-5 minutes on each side.

Remove the patties and cover to keep warm. Sauté the minced garlic in the same skillet for 30 seconds.

Add the tomato paste and cook for 1 minute. Gradually stir in the heavy cream and heat for 2-3 minutes.

Stir in the crumbled feta and smoked paprika, cooking until the sauce thickens.

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

Remove from heat, plate the burgers, and garnish with parsley and more feta if desired.