Cream butter for 1 minute using a stand mixer of hand mixer. Add sugar, and cream for 3 minutes, until light and fluffy.
Add extracts and egg, and mix on low (speed 3 of my Kitchenaid) until combined, scraping the sides down a few times as you go.
In a separate bowl whisk together gf flour, cornstarch, salt, and cream of tartar.
With mixer on low speed, gradually add the flour mixture. Once the flour mixture has been absorbed, turn the speed to medium and beat for 1 minute, until dough comes together and resembles playdough.
Cover bowl and set aside to let dough rest while preheating the oven to 375 degrees F.
Once oven is preheated, place a large piece of parchment paper on your counter and tape it to the counter. Break dough in half and work each half together into a ball. Set in the middle of the parchment. Sprinkle top of dough with gf all purpose flour, and dust your rolling pin with gf flour. Roll dough out to 1/8 to 1/4 inch thick. (Dough may seem like it's breaking a part a bit, but be patient. As you roll more, it will flow together.
Cut out shapes and use a small offset spatula to get under the dough and remove the cut outs. Place on a parchment lined baking sheet. These cookies will not spread so you can put a dozen on a sheet. Bake for 9-10 minutes until set. Remove from oven and let fully cool.
For Mango Buttercream Frosting: Puree mango chunks in a food processor until smooth.
Cream softened butter for 2 minutes in a stand mixer or with a hand mixer, until fluffy and airy. Add powdered sugar and mix for 2 minutes until very creamy. Then add 1/3 cup mango puree, fresh lemon juice, and salt. Beat for 1 minute until fully combined and creamy. Taste and adjust salt if needed.
Frost cookies with buttercream. And enjoy! Store leftover cookies in a closed container in fridge.