Copycat Lofthouse Sugar Cookies

Name: Copycat Lofthouse Sugar Cookies
Prep time: 20 min
Cook time: 9 min
Total time: 29 min
Cuisine: American
Ingredients: 1 cup softened butter (no substitute); 3/4 cup oil; 1 1/4 cups sugar (plus a 1/4 cup for dipping glass into); 2 tablespoons water; 2 teaspoons vanilla; 2 eggs; 1/2 teaspoon baking soda; 1/2 teaspoon Cream of Tartar; 1/2 teaspoon salt; 5 1/2 cups all-purpose flour - measured correctly; 4 ounces cream cheese; 1/4 cup butter, softened; 1 teaspoon vanilla; 2 cups powdered sugar, sifted; 2 teaspoons milk