Irish butter has a high butterfat content, making it especially flavorful. It's expensive to use for everyday baking but worth the splurge for this shortbread, where the ingredient shines.
Prepare an ice-water bath with a medium metal bowl set inside. Melt butter in a medium saucepan over medium-high heat. Continue to cook, stirring constantly, until deep golden brown, about 8 minutes. Immediately pour butter into bowl in ice-water bath and whisk in milk. Stir mixture constantly with a rubber spatula, scraping the sides of the bowl, until mixture begins to thicken, about 3 minutes.
Remove bowl from ice-water bath and let stand until mixture thickens to the consistency of room-temperature butter, about 30 minutes.
Beat butter mixture with a mixer on medium-high speed until pale and fluffy, about 2 minutes. Gradually beat in confectioners' sugar and vanilla, and continue to beat for 2 minutes more. Reduce speed to low; beat in flour.
Butter a 10-inch fluted tart pan with a removable bottom. Press dough evenly into bottom of pan using plastic wrap. (Make sure dough is flat; a small offset spatula may help.) Refrigerate, covered with plastic wrap, for 20 minutes.
Preheat oven to 300 degrees. Cut through dough to make 8 wedges using a sharp paring knife and a ruler. Cut a round from the center using a 2 1/4-inch ovenproof cookie cutter; discard dough. Return cutter to hole. Prick wedges all over with a wooden skewer. Refrigerate, covered, for 20 minutes.
Sprinkle sanding sugar evenly over shortbread. Bake, rotating pan halfway through, until golden brown, 55 minutes to 1 hour 5 minutes. Let cool in pan set on a wire rack for 5 minutes. Remove shortbread from pan, and cut along lines into wedges. Let cool completely on rack.
Shortbread can be stored in an airtight container for up to 2 weeks.