Start by preheating your oven to 350°F (175°C). In a large mixing bowl, combine 1 1/2 cups of softened butter and 1 1/2 cups of granulated sugar. Use an electric mixer to cream them together until the mixture is light and fluffy.
Next, add the 3 large eggs, one at a time, ensuring each is fully incorporated before adding the next. Then, mix in 1 1/2 tablespoons of vanilla extract.
In a separate bowl, whisk together 4 1/2 cups of all-purpose flour, 4 teaspoons of baking powder, and 1/2 teaspoon of kosher salt. Gradually add this dry mixture to the wet ingredients, mixing until just combined.
Using a cookie scoop or tablespoon, drop rounded balls of dough onto lined baking sheets, spacing them about 2 inches apart. Bake for 8-10 minutes or until the edges are lightly golden but the centers are still soft.
While the cookies cool, prepare the frosting. In a mixing bowl, beat 1 cup of softened butter until creamy. Gradually add in the 3 cups of sifted confectioners sugar, mixing until combined. Add 1 tablespoon of vanilla bean paste and 3 tablespoons of half and half, mixing until smooth.
Once the cookies are completely cool, spread a generous layer of frosting on each cookie. If desired, finish with a sprinkle of colorful sprinkles.