To make the basil oil, combine the olive oil, 1 cup basil leaves, and 1/4 teaspoon salt in blender. Blend until smooth. Set aside.
Divide the Feta, stuffed grape leaves, cucumber, tomatoes, olives, pepperoncini, and the basil sprigs among 6 plates. Squeeze with lemon juice; drizzle with the basil oil. Add salt and pepper to taste.