Beat the softened butter and powdered sugar in a large bowl for 2 to 3 minutes, until smooth and creamy. Mix in the vanilla.
Whisk the flour and salt together in a separate bowl. Add them to the butter mixture a little at a time. Mix on low speed just until the dough comes together.
Use a spatula to mix the butterscotch chips through the dough.
Shape the dough into a log about 2 inches across. For cookie cutters, shape it into a flat disc instead. Wrap the dough well and chill for 30 minutes, or until firm enough to slice.
Heat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Cut the cold dough log into rounds about ¼ inch thick. If using a disc, roll it to ¼ inch thick and cut out the cookies. Place them on the lined baking sheets, about 2 inches apart. Add a light sprinkle of coarse sugar if you are using it.
Bake for 12 to 15 minutes. Check near the end. The edges should be lightly golden and the centers should still look pale.
Leave the cookies on the baking sheet for about 5 minutes. Then move them to a wire rack and let them cool completely before serving.