Preheat oven to 400°F. Line at least 2 baking sheets with parchment paper.
In the bowl of a stand mixer using a paddle attachment, beat together butter, sugar, salt, baking powder, lemon zest, lemon extract, and vanilla until well combined. Beat in eggs one at a time, mixing well after each addition. Stir in flour alternately with milk on low speed, beginning and ending with flour until just blended.
Using a large ice cream scoop or a 1/4 cup measure, portion the dough 2 1/2 inches apart on the prepared baking sheets. Have a small cup of water ready to hand. Dip fingers in water and pat the cookie dough mounds to a 3" round.
Bake cookies for 10-12 minutes until set and lightly crackled on top. Remove cookies from the oven and cool on the baking sheet for 5 minutes, or until firm enough to transfer to wire cooling racks. Cool completely.
For the glaze, whisk together confectioner's sugar, corn syrup, hot water and vanilla extract in a large mixing bowl. Mixture will be thick. Divide the icing mixture evenly between three separate bowls. Reserve one bowl of white icing. Stir red food coloring into one bowl and green food coloring into another bowl and mix until food color is thoroughly blended.
Place each glaze in a separate disposable piping bag with the ends snipped. Pipe the tops of each cookie with each colored glaze, one half at a time.
Place cookies on cooling racks and allow them to set at least 30 minutes. Wrap cookies individually in plastic wrap or in cellophane bags and store at room temperature for up to a week.