In a stand mixer, beat together the butter, oil, sugar and powdered sugar until light and fluffy using a whisk attachment.
Add in the vanilla, whisk to combine.
In a small bowl, combine the flour, cream of tartar, baking soda, guar gum, and salt.
Turn the stand mixer to low, and 1 cup at a time, add the flour mixture. Mix until completely combined.
Separate the dough into 2 parts and form into round disks on parchment paper. Wrap individually and chill the dough for one hour.
Preheat the oven to 350 degrees.
Working with one disk at a time, cut the dough into 20 squares, roll the squares into balls, and using your three middle fingers, press the balls onto a parchment lined baking sheet.
Bake for 8-10 minutes. Let the cookies cool on the baking sheet for 2 minutes, then gently, using a spatula transfer them to a cooling rack.
In a small bowl, whisk together the lemon juice, powdered sugar and vanilla to form the icing. The mixture will resemble a glaze.
Once the cookies have cooled. Use a spoon or butter knife to gently spread the glaze onto the cookie. Place the cookies back onto the cooling rack to let the glaze set.