Set out 3 sticks of butter to soften at room temperature. In a large bowl, use a mixer to blend the softened butter with the 1 ¼ cups granulated sugar until creamy. Add eggs and vanilla; mix.
In a separate medium bowl, use a fork to toss together the flour, baking powder and salt. Add some of the flour mixture to the butter mixture, mixing well. Add the half and half and mix. Gradually add the rest of the flour mixture in, beating well after each addition. You will end up with a sticky dough.
Cover the bowl of dough with plastic wrap and refrigerate for 30 minutes. During that time, prepare the glaze. In a small saucepan (stainless steel if you have one), whisk the granulated sugar, water and corn syrup together. Cook over medium heat, stirring until the mixture just comes to a boil. Remove the pan from the heat and stir in the vanilla. Let the mixture cool for five minutes.
After five minutes, use a mixer on low speed to gradually blend in the confectioners sugar until smooth. (If you used a stainless steel pan, you can do this right in your pan. Otherwise, transfer the glaze to a mixing bowl first.)
Preheat the oven to 350 degrees F. Line two (four, if you have them) sheet pans with parchment paper (or lightly grease the pans with cooking spray).
Place a few tablespoons of flour in a small bowl (for dusting your hands). When the dough is chilled, use two teaspoons or a scoop to measure out some dough. With floured hands, roll the dough into a 2-inch ball and place it on the baking sheet. Continue scooping and rolling, dusting your hands each time. Place the cookies two inches apart on the pan. Refrigerate the dough whenever it gets too soft to work with.
Bake the cookies at 350 degrees F for 10-15 minutes, until the tops are set, and the bottoms are lightly browned. (Mine took 13 minutes.) Let the cookies rest for a couple of minutes before placing them on a cooling rack.
When the cookies are completely cooled, you can glaze them. One at a time, dip the cookie in the glaze, top down, and give it a twist. Use your finger to spread the glaze around the surface of the cookie. Immediately sprinkle it with nonpareils.
Place the glazed cookies on a cooling rack and set it over a parchment-lined or wax-paper-lined pan to catch the drips. It will take about 4 hours for the glaze to fully dry, so wait to store or package the cookies.
Once the glaze has dried, store the cookies covered at room temperature for up to a week. Note that if you use a plastic container, the cookies will become very moist. I like to keep mine on a tray loosely covered with plastic wrap. Any leftover glaze can be kept at room temperature for three days, then refrigerated for a week. You also could freeze the glaze. You can freeze unglazed cookies for up to three months, then thaw completely and glaze them.
It will take four hours for the glaze to fully dry on the cookies, so plan ahead if you need to package up the cookies to take or send somewhere. (You'll want them dry before you attempt to stack them.)The cookie dough is very soft and sticky. You will need to refrigerate it for 30 minutes first, then again for several minutes while some of the cookies are baking. Make sure your hands are floured when rolling.If you need to reuse your baking pans as you go, make sure they are cool first, so the cookies don't spread. It helps to rinse the pans in cold water and line them with fresh parchment paper.