Divide beef into 4 equal portions, shaping each into a 1/2-inch-thick patty. Sprinkle patties evenly with 1/4 teaspoon salt and 1/4 teaspoon pepper. Heat 2 teaspoons oil in a large skillet over medium-high heat; swirl to coat. Add patties; cook 3 minutes on each side or until browned; remove from pan.
Add remaining 1 teaspoon oil to pan; swirl to coat. Add vertically sliced onion; sauté 3 minutes or until lightly browned and tender. Sprinkle onion with flour; cook 30 seconds, stirring constantly. Add stock and remaining salt and pepper; bring to a boil. Return patties to pan; reduce heat to medium, partially cover, and cook 5 minutes or until gravy is slightly thickened. Sprinkle with green onions, if desired.