Line two large rimmed baking sheets with parchment paper or silicone baking mats. Clear enough refrigerator space to chill the portioned cookie dough.

Place the softened butter, brown sugar, and granulated sugar in a large mixing bowl. Beat on medium speed for 2 to 3 minutes, until the mixture looks pale, fluffy, and evenly blended. Scrape down the bowl once during mixing.

Add the eggs one at a time, beating on low speed until each is fully incorporated. Mix in the vanilla. The mixture should look smooth and creamy rather than separated or greasy.

Add the dry instant butterscotch pudding mix: Beat on low speed for about 30 seconds, until no dry streaks remain and the mixture is evenly golden.

In a medium bowl, whisk together the flour, baking soda, and salt. Break up any flour lumps so the baking soda is distributed evenly.

Add the dry ingredients to the butter mixture in two additions, mixing on low speed after each. Stop when only a few flour streaks remain, then finish folding by hand. The dough should be thick, soft, and slightly tacky but firm enough to scoop.

Fold in the butterscotch chips until they are evenly distributed. Avoid prolonged mixing, which can make the cookies tougher.

Scoop the dough into 24 portions of about 2 tablespoons each. Arrange them on the prepared baking sheets, leaving about 2 inches (5 cm) between portions. Roll any uneven scoops gently between your palms without compacting them.

Near the end of the chilling time, preheat the oven to 350°F (175°C). Position a rack in the center of the oven.

Bake one sheet at a time for 10 to 12 minutes. The edges should look set and lightly golden, while the centers should appear puffed and slightly underdone. The cookies should not be deeply browned across the tops.

If the cookies have spread unevenly, immediately place a round cookie cutter or wide glass around each warm cookie and gently move it in a circular motion to neaten the edges.

Let the cookies cool on the baking sheet for 5 minutes. They will be very soft when first removed from the oven but will firm as they rest.

Bake the second sheet for 10 to 12 minutes. Transfer the first batch to a wire rack, then repeat the 5-minute pan cooling process with the second batch.

Move all cookies to a wire rack and let them cool for at least 5 more minutes before serving. The edges should be lightly crisp, with soft and chewy centers.