Preheat oven to 375°F (190°C).
In a large bowl, cream together the softened butter and white sugar until light and fluffy.
Beat in the eggs one at a time, then mix in vanilla extract.
In another bowl, whisk together flour, baking powder, and salt.
Gradually add dry ingredients to the wet ingredients, mixing until fully combined.
Divide the dough in half, wrap each portion in plastic wrap, and refrigerate for at least 2 hours.
Roll out one portion of dough on a floured surface to ¼-inch thickness.
Cut into shapes with cookie cutters and place on ungreased or parchment-lined baking sheets.
Bake for 8–10 minutes, or until edges are just beginning to brown.
Cool on baking sheets for 5 minutes, then transfer to wire racks to cool completely.
For the icing, stir together powdered sugar, milk, honey or corn syrup, and vanilla until smooth. Adjust milk as needed for desired consistency.
Add food coloring if desired.
Decorate cooled cookies and allow icing to set before serving or storing.