Preheat the oven to 350°F (175°C) and line two large baking sheets with parchment paper.

In a stand mixer fitted with the paddle attachment, cream 1 cup softened butter and 1 1/4 cups granulated sugar on medium-high speed for 3 minutes until light and fluffy.

Add eggs one at a time, beating well after each addition, then incorporate 1 tablespoon vanilla extract.

In a separate bowl, whisk together the flour, cream of tartar, baking soda, and salt.

Gradually add the dry ingredients to the wet mixture on low speed until just combined, ensuring not to overmix the dough.

In a small bowl, combine the 1/4 cup sugar and 1 tablespoon cinnamon for the coating.

Divide the dough into 12 equal portions (approximately 1/3 cup each) and roll into smooth balls.

Thoroughly coat each ball in the cinnamon-sugar mixture and place them on the prepared baking sheets, spaced 3 inches apart.

Bake for 10-12 minutes until the edges are set and the tops are slightly puffed; the centers should still appear soft.

Cool the cookies on the baking sheet for at least 15 minutes before transferring to a wire rack to cool completely.

To prepare the frosting, beat the cream cheese and 1/2 cup butter until smooth, then gradually add powdered sugar and 1 teaspoon vanilla, whipping until stiff peaks form.

Using a piping bag with a large round tip, pipe a spiral swirl starting from the center of each cooled cookie moving outwards.