Microwave butterscotch chips and 3 tablespoons butter in a microwave-proof bowl until melted, about 1 minute, stirring halfway through. Whisk in vanilla until mixture is smooth; let cool for 10 minutes.
Cut remaining 9 tablespoons butter into 9 pieces. Using a stand mixer fitted with the paddle or beater, beat remaining 9 tablespoons butter, sugar, and salt on medium-high speed until pale and fluffy, about 3 minutes. Add cooled butterscotch mixture and beat until thoroughly combined, about 1 minute, scraping down bowl as needed. Add egg yolk and beat until incorporated, about 30 seconds. Reduce speed to low, add flour and baking powder, and mix until dough forms, about 1 minute.
Turn out dough onto counter and roll into a 9-inch log. Wrap tightly with parchment paper and roll to form a round cylinder (good luck with that... as you can see from the photos, mine were quite oblong). Refrigerate until firm, at least 2 hours or up to 3 days.
Adjust oven racks to upper-middle and lower-middle positions and heat oven to 325°F. Line 2 baking sheets with parchment paper. Slice chilled dough with a sharp knife into ¼-inch-thick rounds and place 1 inch apart on prepared baking sheets (16 cookies per sheet - I needed 3 sheets because I ended up with 37 cookies). If the chilled dough begins to split as you slice it, soften it on the counter for 10 minutes.
Bake until edges have darkened slightly, about 15 minutes, switching and rotating sheets halfway through baking. Let cool for 10 minutes on baking sheets, then transfer cookies to wire rack to cool completely. Store cookies in an airtight container at room temperature.
Source: America's Test Kitchen