In a large bowl with a hand mixer (or stand mixer), cream together butter and sugar. Add eggs and maple extract and beat until fluffy, about a minute.
In a separate bowl, sift (or whisk) together the flour, baking powder, and salt.
Add dry ingredients to butter mixture and mix until just combined. Shape dough into two round discs and wrap in wax paper. Refrigerate for 1 hour, or up to 24 hours (dough can be made a day ahead of time).
Preheat oven to 350 degrees Fahrenheit and line a baking sheet with parchment paper. Remove one disc of dough at a time from the refrigerator.
Dust counter top, rolling pin, and top of dough with flour. Roll dough out to ¼" thickness, dusting with additional flour as needed to prevent sticking. Using cookie cutters, cut into various shapes and place onto prepared baking sheet about 1 inch apart.
Bake cookies for 8-9 minutes or until cookies just begin to lightly brown around edges. Remove from oven and allow to cool for 5 minutes on baking sheet before transferring to a cooling rack to cool completely.
In a bowl, combine confectioner's sugar and maple syrup, using a whisk if needed to remove clumps. Glaze should be smooth and shiny.
Using a small spatula, the back of a spoon, or a butter knife, gently coat each cookie with a thin layer of glaze.
Allow cookies to dry for several hours before stacking and/or storing.
Your cookies will keep for 5-7 days in a sealed container. To help keep cookies fresh, you can place a slice of bread in the container with the cookies. Replace the bread slice as needed.
Dough scraps can be reformed and rerolled once before additional refrigeration is needed. Warm dough will spread more in the oven and cookies may lose their shape. If your cookies are spreading in the oven - after cutting out your cookies, place entire baking sheet in the refrigerator for 10 minutes just before baking. This will help your cookies keep their shape.
Leftover dough and leftover baked cookies (that have been cooled completely) can be frozen in a tightly sealed freezer-safe container for up to 3 months.