Prep. Preheat the oven to 350°F with the rack positioned in the center. Prep two cookie sheets with parchment paper or a light misting of cooking spray.
Cream butter, sugar, egg and vanilla. In a large mixing bowl, beat the butter, both sugars, egg, and vanilla with an electric mixer on medium speed. The mixture should be light and fluffy after about 3-5 minutes. Scrape down the sides of the bowl when necessary.
Dry Ingredients. In a separate bowl, whisk together the flour, rolled oats, baking soda, and salt. Gradually integrate this dry mixture into the butter mixture on a low speed, adding about a cup at a time. Once combined, scrape down the sides of the bowl. and gently fold in the butterscotch chips.
Add butterscotch chips. Gently fold in the butterscotch chips, mixing by hand or on low speed.
Shape. Using a tablespoon, portion out the cookie dough onto the prepared sheets, aiming for 12-15 cookies per sheet.
Bake. Bake in the preheated oven for about 10-12 minutes. The cookies are ready when the edges have adopted a golden brown hue. Let them cool on wire racks.