Sift the flour, baking powder, and salt into a large bowl. Whisk well, to mix.
Combine the butter and shortening in a separate large bowl. Using an electric mixer, beat the ingredients until smooth and creamy, gradually adding the sugar. Beat in the eggs, one at a time, beating well after each addition. Blend in the vanilla. Using a wooden spoon, stir the dry ingredients into the creamed mixture about 1 cup at a time.
When all of the flour has been added, the dough will be cohesive and semi-firm, but pliable. Turn the dough out onto a lightly floured surface. Divide it into thirds.
To make rolled cookies, place each third of the dough in the center of 1 gallon plastic storage bag and flatten with your palm to about a 3/4-inch thickness. Refrigerate for at least 4 hours, or overnight.
Remove each portion from the fridge about an hour before baking and roll the dough right in the bag to about a 1/2-inch thickness.
Preheat the oven to 350°F. Lightly butter two large baking sheets or line them with parchment paper.
Remove the dough from the bag and, on a sheet of lightly floured waxed or parchment paper, roll it to a little less than 1/4-inch thick. Cut the dough into shapes. Place the shapes on the prepared baking sheet, leaving about an inch in between. (Gather the scraps, press them together until they’re about 3/4-inch thick, place in a plastic bag, and chill until firm, 1 to 2 hours. Roll and cut as above.) Brush lightly with milk and sprinkle with sugar.
To make sliced cookies, place each third of the dough on a sheet of plastic wrap. By folding the wrap over the dough and then rolling it while twisting and lightly pulling on the ends of the wrap to apply tension, shape the dough into logs about 1-1/2 inches in diameter. Refrigerate the dough for at least 4 hours, or overnight.
Remove each log from the fridge about an hour before baking, remove the plastic, and slice the dough into rounds about 1/4-inch thick.
Preheat the oven to 350°F. Place the rounds on prepared baking sheets. Brush lightly with milk and sprinkle with sugar.
For either method, bake on the center oven rack for 12 to 13 minutes, one sheet at a time, until the cookies start to turn golden around the edges. Using a spatula, immediately transfer the cookies to a cooling rack and cool thoroughly. Makes 24 or more cookies, depending on their size.