Line 2 baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the shortening, sugar, eggs, milk, and vanilla extract together until smooth, about 1-2 minutes.
1 cup vegetable shortening, 1 cup granulated sugar, 2 large eggs, 2 tablespoons milk, 1 tablespoon pure vanilla extract
In a separate medium bowl, whisk the flour, baking powder, and salt together.
3½ cups all-purpose flour, 2 teaspoons baking powder, 1 teaspoon kosher salt
Add the dry ingredients to the wet ingredients and mix just until well combined and the dough starts to stick together, about 1-2 minutes. Do not overmix.
Sprinkle a clean work surface lightly with flour and place about ⅓ of the dough onto the prepared surface to roll out. Sprinkle the top of the dough with flour to prevent the dough from sticking to the rolling pin. Roll out the dough to ⅜-inch thickness and cut out cookies into 3½-inch squares.
Place the cut out cookies onto 1 prepared baking sheet. The cookies can be placed close together on the baking sheet for this step. Place the cookies in the freezer for 20 minutes before baking.
While the cookies chill, preheat oven to 350°F
Place 6 cookies onto the second baking sheet about 2 inches apart and bake for 12-14 minutes for thick cut cookies or 8-10 minutes for thinner cookies. The tops of the cookies shoud look set (no longer wet but not yet brown), and the bottoms of the cookies should be slightly brown. The middle of cookies will still be slightly soft to the touch but will continue baking once removed from the oven.
Repeat with remaining cookies. Let cookies cool completely before frosting.