In a mixing bowl and using an electric hand mixer, blend together the all-purpose flour, ground cinnamon, baking powder, and baking soda. Set this bowl aside.
With your mixer on low speed, blend together the butter, brown sugar, and salt until smooth. Scrape down both the bowl and the beaters.Once the butter is smooth, add the egg, vanilla, and rum (or rum extract) to the bowl. Blend, once more, on low speed until the egg is fully incorporated into the butter.
Add half of the dry (flour) mixture to the bowl, and blend on low speed to mix it into the butter.Scrape down the bowl and the beaters.
Next, add all of the coquito to the batter. Blend on low for 30 seconds. The dough may look grainy at this point. Add the last of the flour and blend on low speed for one minute. The dough should look like a very thick frosting.
Cover the bowl with plastic wrap and chill it in the fridge for 45 minutes to 1 hour.
After chilling and 15 minutes before you plan to start baking, preheat your oven to 350°F (177°C). Line two or three sheet pans with silicone baking mats or parchment paper.
Use a number 30 portion scoop, or a rounded tablespoon, to scoop the cookies into balls. Deposit 12 of these balls of cookie dough onto the prepared sheet pan.Continue scooping the cookies until you have scooped all of the dough.If you have more than one sheet pan, you can scoop all of the cookie dough and refrigerate the balls of dough while you begin baking the first pan of cookies.
Bake the cookies for 15 minutes. Remove the pan of cookies from the oven and allow them to cool on the sheet pan for 1-2 minutes. Use a spatula to transfer the cookies to a cooling rack, then allow them to cool completely.
Buttercream:
With your mixer on low speed, blend together the powdered sugar, butter, and salt in a mixing bowl. Because there's more sugar than butter, the mixture may resemble white sand. That's the way it should look.
Pour the coquito and vanilla into the sugar and blend again at low speed. Stop the mixer and scrape down the bowl and beaters. Continue mixing on low until the buttercream is fluffy and evenly mixed.
If you omitted the coquito, thin the buttercream with whole milk, instead. Just add a tablespoon at a time until it reaches a thick but spreadable consistency.Once the coquito buttercream is mixed, covered it with plastic wrap, and set it aside or in the fridge until you're ready to frost your cookies.
Scoop a generous amount (about 1 ½ tablespoons) of the coquito buttercream onto the center of each cookie.
Use an offset spatula or the back of a spoon to spread the buttercream into a circle.
Note: Sprinkle a generous pinch of lightly toasted coconut flakes and/or a sprinkle of ground cinnamon over the coquito frosting. Serve them with coquito, milk, tea, or coffee.
