Beat the butter and sweetener together on medium-high speed 1 minute until creamy Scrape down the sides of the bowl. Beat in the eggs and the extract on medium-high until fully combined. Set aside. In separate bowl whisk together the dry ingredients until there are no lumps.
Turn mixer on to low and gradually add the dry ingredients to the wet ingredients until fully combined. Once combined beat on medium-high until a smooth dough form. Scrape down the sides of the bowl and form the dough into a smooth ball. Place dough back into the mixer bowl, place in the refrigerator uncovered and let chill 7 to 10 minutes until firm.
While the dough is chilling preheat an air fryer to 325 degrees or an oven to 350 degrees. Remove from the refrigerator. Line a clean worksurface with parchment paper. Scoop the dough out 1 TBS at a time onto the worksurface and roll each scoop into a smooth ball.
Use hands and slightly flatten the tops of the dough balls. Do not press too thin. Line the preheated air fryer basket with parchment paper (NEVER PUT LOOSE PARCHMENT PAPER IN AN AIR FRYER.) Place cookies in the air fryer. Leave space between each cookie. If using oven place cookies on lined baking sheet with space between cookies.
Air fry at 325 degrees for 5 to 7 minutes or until golden. If baking, bake at 350 degrees 10 to 12 minutes or until golden. Open air fryer/oven. Allow the cookies to sit in the air fryer basket at least 5 minutes to firm up. Remove from the fryer/baking sheet. Place the cookies on a wire rack and allow to cool completely before applying the icing.