In the mixing bowl of a stand mixer, place the butter, oil, sugar, powdered sugar, water, lemon juice, lemon zest, eggs, and several drops of yellow food coloring gel Mix until all ingredients are blended. In a separate bowl, mix the flour, baking soda, cream of tartar, and salt. Add flour mixture to mixing bowl, in two or three additions, blending between each addition. Blend until dough is smooth.
Line a cookie sheet with parchment paper, and set the oven to 350 Degrees. Roll the cookies into Tablespoon size balls, and roll them in sugar, and place them on the cookie sheet. Smash the cookies, gently, with the bottom of a glass, and place them in the oven for 12 - 13 minutes or until cookies are lightly golden brown around the edges.
Remove cookies to a wire rack to cool completely. While cookies are cooling, make the Icing. In the mixing bowl of a stand mixer, place the softened cream cheese, butter, lemon juice, lemon zest, vanilla, and blend until smooth and creamy. Add powdered sugar a little at a time, until all the powdered sugar has been added, blending between additions.
When cookies are completely cool, fill the center of each cookie with the cream cheese Icing, and sprinkle with lemon zest. Allow cookies to sit for about an hour for Icing to set.
Serve and Enjoy!