Preheat oven to 350F. Line two medium sheet pans with parchment paper and set aside.
Place butter, 2/3 cup sugar, and molasses in the bowl of a large stand mixer, fitted with a paddle attachment OR in a large mixing bowl with a wooden spoon. Mix on medium speed or by hand until creamy, about 2 minutes. Add egg, ginger, and vanilla, beating well until smooth.
In a separate medium bowl, combine flour, cream of tartar, baking soda, ginger, and salt. Fold into butter mixture until dough is just combined.
In a small bowl, combine remaining 1/4 cup sugar and cinnamon.
Use a medium cookie scoop to scoop dough out into 1-in balls. Roll balls in cinnamon sugar until evenly coated and place 2 inches apart on prepared sheet pans. Flatten cookies very slightly with the palm of your hand.
Bake cookies at 350F 9-10 minutes until edges of cookies are golden brown. Allow cookies to cool on sheet pans 10 minutes before transferring to a wire cooling rack to cool completely. Enjoy!