In an electric mixer bowl, blend the sugar, butter, and egg. Add the lemon extract and beat for 1 minute.
Sift the flour, baking powder, and salt onto a square of wax paper (or mix in a bowl).
Add the dry ingredients alternately with the milk to the butter mixture, mixing well after each addition. Divide the dough in half, wrap in plastic, and refrigerate until firm, at least 1 hour.
Preheat the oven to 400 degrees. On a lightly floured surface, roll half the dough to 1/4 inch thick. Dip a cookie cutter into flour, cut into the dough, and place the cookies on an ungreased cookie sheet. Repeat with the other half of the dough. Scrape the dough scraps together, add flour to the rolling surface as needed, roll out again, and cut out additional cookies. Bake 7 to 10 minutes and cool on racks.
To make the icing, combine the butter, powdered sugar, and vanilla in a mixer bowl. Beat with an electric mixer until blended. Add the evaporated milk 1 tablespoon at a time and continue to beat to desired spreading consistency. If desired, divide the icing into separate bowls and tint each one a different color, starting with just a drop of food coloring.