Combine 1 cup unsalted butter, room temperature and 3/4 cup confectioners' sugar in a large bowl. Beat on medium speed with an electric mixer until fluffy. Add 1 tablespoon vanilla extract and beat until combined.
Whisk 1 teaspoon baking powder and 2 cups all-purpose unbleached flour together in a separate bowl. Add this flour mixture slowly to the butter mixture, stirring on a low speed. Stop beating as soon as the flour mixture is completely incorporated. Over-beating will result in toughness.
Form the dough into a slightly flattened ball and chill for about 30 minutes. Place chilled dough into the plastic zipper storage bag. With the bag slightly open to allow air to escape, roll out the dough until evenly distributed throughout the bag. Place back into the refrigerator until firm--at least 30 minutes--up to 2 days. It can also freeze at this point.
Preheat the oven to 350˚F (180˚C). Line baking pans with parchment paper, silicone mats, or use cooking spray.
Remove cookie dough from the fridge and carefully cut three sides of the bag with kitchen shears or scissors, leaving one side intact. Open up and use cookie cutters to cut out cookies. Because the dough is cold and firm, they should peel off the plastic easily. Place on prepared cookie sheets. Re-roll scraps after replacing plastic over the new ball and refrigerate for a few minutes until firm again. Repeat as needed until all dough is used.
Bake cookies for 10 to 12 minutes, until light brown around the edges. Cool the cookies for 10 minutes on the baking sheet. Remove and cool completely on a wire rack before icing.
Frosting
Whisk together 2 cups powdered sugar, 3-4 tablespoons milk, and 1/2 teaspoon vanilla or almond extract. If too thick, add more milk, a few drops at a time. If too thin, add more powdered sugar or let sit for a while.
Use 1-2 drops pink food coloring to tint the icing. Ice cookies and allow them to sit until the icing dries. Store cookies in an airtight container at room temperature for up to 1 week or tightly wrapped in aluminum foil or plastic for one month in the freezer.