In a small bowl, whisk together the 1/3 cup granulated sugar and 1 teaspoon chai spice. Set aside.

Preheat oven to 375°F. Line a large baking sheet with parchment paper.

In a medium bowl, whisk together the flour, cream of tartar, baking soda, salt, 1 teaspoon chai spice, and cinnamon.

In a large bowl using a hand mixer or stand mixer with paddle attachment, beat the softened butter and 1 1/3 cups granulated sugar on high speed until light and creamy, about 2 minutes. Add the eggs and vanilla extract, beating on medium-high until well combined.

With the mixer on low speed, gradually add the dry ingredients to the wet mixture, mixing just until a soft dough forms. Do not overmix.

Scoop the dough into balls, using about 2 tablespoons of dough each. Roll each ball in the prepared spiced sugar topping to coat evenly. Place the coated dough balls on the prepared baking sheet, spacing them about 3 inches apart.

Bake for 10 to 14 minutes, or until the edges are set and lightly golden. Allow the cookies to cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely before frosting.

In a stand mixer fitted with the whisk attachment, beat the softened butter, powdered sugar, vanilla extract, and 1/4 teaspoon chai spice on low speed until combined. Gradually add the heavy cream, one tablespoon at a time, beating on medium-high until the frosting is light, fluffy, and spreadable. Adjust cream as needed for desired consistency.

Once the cookies are completely cool, frost each cookie with the chai latte buttercream. Serve immediately or store in an airtight container.