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, then set aside for about 5 minutes.

Divide the mixture into four equal portions and shape into round, 1-inch thick patties. Make a small indentation in each patty's center.

Heat olive oil in a large skillet over medium-high heat. Cook patties for 4-5 minutes on each side until golden-brown and reach 160°F for medium.

For the sauce, reduce heat and sauté minced garlic until fragrant. Stir in tomato paste, then gradually pour in heavy cream, mixing well.

Add crumbled feta and smoked paprika, seasoning with salt and pepper. Return cooked patties to skillet and spoon sauce over them, heating for 1-2 minutes.

Plate the burgers with the sauce drizzled on top, garnished with parsley and additional feta crumbles if desired.