In a medium bowl, cream the butter, then gradually add the sugar and continue to beat. Add the eggs, vanilla extract, maple extract, and maple syrup, and beat until light and fluffy.

Sift the flour, baking powder and salt together. Thoroughly blend into the butter mixture. Form into a disc, wrap in plastic wrap, and chill for at least 4 hours.

Preheat the oven to 375 degrees F.

Roll the dough out 1/4-inch thick. With a drinking glass, cut out cookies and transfer to parchment lined baking sheet. Brush the cookies with heavy cream. Bake for 8 to 10 minutes.

Meanwhile make the maple glaze. Mix all the ingredients in a saucepan. Stir over medium heat until it just reaches the boiling point. Brush or dip the tops of the cookies in the maple glaze while still hot. Let cool. These taste better the next day.