Set the oven to 350°F. Let it reach temperature before the first tray goes in.
Combine the flour, baking powder, baking soda, and salt in a medium bowl with a whisk, then reserve. The mixture should look evenly blended.
Beat both sugars with the softened butter in a large bowl on high speed for 2 to 3 minutes using an electric mixer or a stand mixer with a paddle. Stop when the mixture looks lighter and fluffy.
Beat in the yolks, vanilla, butter extract, and cooled melted chips on medium speed for 1 to 2 minutes, scraping the bowl as needed. The mixture should become pale and aerated, without hot chips melting the butter.
Work in the flour mixture on low speed, stopping as soon as it comes together. No dry flour should remain visible.
Distribute the whole butterscotch chips through the dough with a rubber spatula. The chips should appear throughout the mixture.
Portion the dough into 2-tablespoon scoops with a large cookie scoop. Keep the portions consistent for even baking.
Place six portions on a parchment-lined baking sheet, leaving 2 inches between them, and bake for 10 to 11 minutes. The centers may still look slightly soft.
Immediately circle each hot cookie with a large round cookie cutter to nudge its edges inward. The warm edges should settle into a rounded shape.
Leave the cookies on the sheet for 5 minutes before moving them to a wire rack to cool completely. The centers firm as the cookies rest on the warm pan.