Preheat the oven to 350 degrees F (175 degrees C).
Beat butter in a bowl with an electric mixer until creamy. Add 1 cup sugar, vegetables oil, lemon juice, and vanilla extract; mix to combine. Add egg and mix until thoroughly incorporated.
Sift together flour, cream of tartar, baking soda, and salt in a bowl. Add flour mixture to the butter mixture; mix until well combined.
Pour 1/4 cup sugar into a small bowl. Form dough into 1-inch balls. Roll each ball into sugar to coat and place on an ungreased baking sheet.
Bake in the preheated oven until lightly browned, about 10 minutes. Remove from oven, transfer to a wire rack and cool completely, about 30 minutes.
Mix together powdered sugar and lemon juice for the frosting in a small bowl until smooth. Pour frosting over cookies and allow to harden for at least 5 minutes before serving.
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