Preheat oven to 350°F (175°C) and line baking sheets with parchment paper.
In the bowl of a stand mixer fitted with the paddle (or using a hand mixer), cream 1 1/2 cups softened butter with 1 1/2 cups granulated sugar until light and fluffy, about 3 minutes.
Add the eggs one at a time, mixing well after each addition; then increase the mixer speed and mix in 1 1/2 tablespoons vanilla extract.
In a large bowl whisk together 4 1/2 cups all-purpose flour, 4 teaspoons baking powder, and 1/2 teaspoon kosher salt.
With the mixer on low, gradually add the dry ingredients to the butter mixture and mix until just combined into a soft dough.
Scoop about 2 tablespoons of dough per cookie and place on prepared baking sheets, spacing evenly; gently press each mound with your palm to slightly flatten.
Bake cookies in the preheated oven for 10 minutes, or until edges are set but centers remain soft; do not overbake.
Remove cookies from oven and allow them to cool on the baking sheet while you prepare the frosting.
For a cookie cake: spray a 9-inch skillet or cake pan with nonstick spray, press the cookie dough evenly into the pan, and bake at 350°F for 12–15 minutes until a toothpick inserted in the center comes out clean; set aside to cool.
To make the frosting, cream 1 cup softened butter in a large bowl. Gradually add the sifted confectioners' sugar 1/2 cup at a time, increasing mixer speed after each addition to incorporate air.
Once all the powdered sugar is incorporated, mix in 1 tablespoon vanilla bean paste, then add the 3 tablespoons half and half one tablespoon at a time until the frosting reaches a spreadable consistency.
Spread the frosting over cooled cookies or the cooled cookie cake, decorate with sprinkles if desired, and serve.