Use electric mixer with paddle attachment to beat butter and sugar until lightened in color, about 2 minutes.
Beat in vanilla extract. Then, beat in eggs, one at a time, scraping down sides as needed, until well incorporated.
In a medium sized bowl, stir together flour, baking powder, salt, and corn starch. Gradually beat into butter mixture, about 1/2 cup at a time.
Form dough into about a large disc, about 8" in diameter. Cover with plastic wrap and chill in refrigerator for 3-4 hours.
Preheat oven to 400 degrees. Line a baking sheet with parchment paper or a silicone baking mat.
Lightly flour a working surface. Take about half of the chilled dough out of the fridge. Roll to between 1/4 and 1/8 inch thickness (I use a ring to roll to 3/16 inch thickness). Cut out as many cookies as possible out of each roll (the less you roll, the better). Repeat with the rest of the dough until all dough has been used. NOTE: If you have trouble rolling the dough, let it sit for a few minutes to warm just a bit.
Transfer cutout cookies to prepared pan and bake until golden brown, about 6-8 minutes.
Let cool on pans for about 2 minutes, then transfer to cooling rack to finish cooling.
I decorate these cookies with the royal frosting recipe from my sugar cookie post (link in notes). I have notes above about decorating in post above.