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

Divide the mixture into four equal portions and shape each into a patty, making a slight indent in the center.

Heat a skillet with olive oil over medium-high heat. Cook patties for 4-5 minutes on each side until browned and cooked to your desired doneness.

Remove cooked patties from the skillet and cover with foil to keep warm.

Lower the heat, add a teaspoon of olive oil if needed, and sauté minced garlic for 30 seconds until fragrant.

Stir in tomato paste and cook for 1 minute. Gradually pour in heavy cream while stirring.

Cook the mixture for 2-3 minutes until slightly thickened. Incorporate crumbled feta and smoked paprika, seasoning with salt and pepper.

Return patties to the skillet and spoon creamy sauce over them, heating for an additional 1-2 minutes.

Plate the burgers, drizzle with extra sauce, and garnish with fresh parsley.