In a large bowl, use a stand mixer or a hand mixer and beat the softened butter and sugar on medium speed until combined and creamy, about 2 minutes.
Add the egg and vanilla extract; mix on high until fully combined. Scrape down the sides and bottom of the bowl as needed using a rubber spatula.
In a separate medium bowl, whisk together the flour, cinnamon, ginger, cardamom, allspice, nutmeg, cloves, baking powder, baking soda, and salt.
With the mixer running on low speed, gradually add the flour mixture to the butter mixture. Use a rubber spatula to scrape down the edges and bottom of the bowl, as needed. Once there are no more flour streaks, stop mixing. Do not over mix the dough. Then, place the dough in the fridge to chill for 30 to 60 minutes.
Preheat oven to 400 degrees F. and prepare a few baking sheets with parchment paper, or line with a silicone baking mat.
To roll the cookies: Combine 1/3 cup sugar with the 1 1/2 tablespoons cinnamon in a small bowl. For each cookie, scoop out about 1 1/2 tablespoons of the dough and roll it into a ball. Roll the dough balls in the cinnamon/sugar mixture and place them 3 inches apart on a cookie sheet.
Bake cookies for 8 minutes and no more. The cookies may seem undercooked, but will continue to set after they are removed from the oven.
Allow cookies to cool on the pan for at least 5 minutes. Then, transfer them to a wire rack to cool completely.
Make the icing: Whisk all of the ingredients together until no lumps remain. If the glaze is too thick, add more milk, and if the glaze is too thin, add more powdered sugar until it has reached your desired consistency. Drizzle a small amount over each cookie.