Preheat the oven to 350℉ (177℃). Line two baking sheets with parchment paper. In a shallow bowl, combine the sugar and cinnamon. Set aside.
⅓ cup (67g) granulated sugar, 1 teaspoon ground cinnamon
In a large mixing bowl, combine the butter, brown sugar, and granulated sugar. Beat on medium-high speed for 2 minutes until creamy and light. Add the eggs and vanilla, mixing again until combined. Stop to scrape the bottom and sides of the bowl.
1 cup (227g) salted butter, ¾ cup (150g) light brown sugar, ½ cup (100g) granulated sugar, 2 large eggs, 2 teaspoons pure vanilla extract
Add the flour, baking powder, baking soda, cinnamon, and salt. Mix on medium-low speed just until combined and no streaks of flour remain.
2 ¾ cups (388g) all-purpose flour, 1 teaspoon baking powder, ½ teaspoon baking soda, 1 teaspoon ground cinnamon, ½ teaspoon salt
Use a large cookie scoop to scoop 1/4 cup portions of dough onto the prepared baking sheets. Roll each mound of dough into a ball between the palms of your hands, then roll them in the cinnamon-sugar mixture. Reserve leftover cinnamon-sugar mixture for later. Gently press the rolled balls of dough into pucks that are roughly 1-inch thick.
Bake for 10-12 minutes until set around the edges. Let cool on the pans for 10 minutes before transferring to a wire rack to cool completely.
To make the frosting, beat the softened butter, powdered sugar, brown sugar, cinnamon, vanilla, and cream in a large mixing bowl using an electric mixer until fluffy and smooth. Transfer the frosting to a piping bag fitted with a round open tip and pipe concentric swirls of frosting onto the cooled cookies. Finish with a sprinkle of leftover cinnamon sugar before the frosting has a chance to set.
12 Tablespoons (170g) salted butter, 3 cups (360g) powdered sugar, ¼ cup (50g) brown sugar, 2 teaspoons ground cinnamon, 2 teaspoons pure vanilla extract, 2-3 Tablespoons heavy cream or milk