In a medium bowl combines the butter, sugar, vanilla, and egg. Use a wooden spoon to thoroughly mix all the ingredients. Next, add the orange zest and orange juice into the butter mixture and mix until creamy and smooth.
Place baking sheet with the unbaked orange sugar cookies in the refrigerator while you preheat the oven to 350°Fahrenheit. Bake on middle rack for 15-18 minutes or until slightly brown around the edges.