In a large mixing bowl, combine ground beef, egg, crumbled feta, basil, garlic, salt, and pepper. Mix gently until just combined—avoid overmixing.
Divide mixture into 14 mini patties (2–3 inches in diameter).
Heat a large skillet over medium-high heat. Cook patties in batches for 3–4 minutes per side, or until cooked to desired doneness. Set aside and keep warm.
Drain excess grease from the skillet, leaving a thin layer.
Lower heat to medium. Add beef stock to deglaze the pan, scraping up the browned bits.
Whisk in heavy cream, tomato paste, Italian seasoning, and green onions. Simmer for 3–5 minutes until thickened. Season to taste.
Return patties to skillet and simmer for 1–2 minutes to meld flavors.
Serve with extra sauce and garnish with chopped green onions.