Preheat the oven to 350°F and cover two baking sheets with parchment paper.
Bring the refrigerated ingredients to room temperature before mixing. Properly softened butter will cream more effectively and help the cookie dough develop a smooth, even consistency.
Sift the 1 cup powdered sugar intended for the frosting. Melt the ¼ cup butterscotch chips for the cookie dough and the separate ½ cup portion for the frosting, keeping them in different bowls. Allow both melted portions to cool slightly.
Combine the all-purpose flour, baking soda, baking powder, and ½ teaspoon salt in a medium bowl. Whisk thoroughly so the leavening ingredients are distributed evenly.
Set the flour mixture aside until needed. Incorporating the dry ingredients only after the wet mixture is complete reduces the amount of stirring required and helps prevent an overly firm cookie texture.
Place the softened ¾ cup butter, packed brown sugar, and granulated sugar in a large mixing bowl.
Beat for 2–3 minutes, until the mixture becomes pale, airy, and noticeably increased in volume. This creaming stage introduces air into the mixture and contributes to the cookies' tender structure.
Add the room-temperature egg yolks, vanilla bean paste, and butter extract to the creamed butter mixture. Beat for approximately 1 minute, or until completely incorporated.
Mix in the cooled ¼ cup melted butterscotch chips until the dough appears uniform in color.
Using a spatula or wooden spoon, gently fold in the prepared dry ingredients. Stop mixing as soon as the flour is no longer visible.
Fold the remaining ½ cup whole butterscotch chips into the dough, distributing them as evenly as possible.
Measure the cookie dough into approximately 2-tablespoon portions. Roll or shape each portion into a ball and place them on the prepared baking sheets, leaving about 2 inches between cookies.
Bake for 10–11 minutes. The outer edges should look set while the centers remain slightly underbaked.
Allow the cookies to rest on the baking sheets for 5 minutes after removing them from the oven. Transfer them to a wire rack and cool completely before decorating.
If a more precise round shape is desired, gently circle each warm cookie with a round cutter immediately after baking.
While the cookies cool, pour the butterscotch beer or cream soda into a medium saucepan. Bring the liquid to a simmer over medium heat.
Continue simmering for 15–17 minutes, stirring occasionally, until the beverage reduces to a thick, syrup-like consistency that coats the back of a spoon.
Turn the heat to low and stir in the 3 tablespoons softened butter until completely melted. Add the room-temperature heavy cream and a pinch of salt, then stir gently for approximately 1 minute.
Remove the saucepan from the heat and allow the butterbeer caramel to reach room temperature before using.
In a clean mixing bowl, beat the softened ½ cup butter together with ⅛ teaspoon salt for 5–10 minutes. Continue whipping until the butter is very light, creamy, and pale.
Add the cooled ½ cup melted butterscotch chips and blend until incorporated.
Gradually mix in the sifted powdered sugar, followed by the butter extract and vanilla bean paste. Beat for approximately 1 minute, or until the frosting becomes smooth and easily spreadable.
After the cookies have cooled completely, pipe or spread a generous amount of butterscotch frosting onto each cookie. Finish by drizzling the cooled butterbeer caramel over the frosting.