Melt butter in a small kettle over medium heat. Once melted, the butter will begin to sizzle and foam. Continue cooking butter until foaming subsides and brown bits appear on the bottom of the kettle. After 5-7 minutes, the butter will turn a deep golden color and the butter will smell toasted ( I promise you will notice the nutty smell of the butter). Set aside and let the butter cool to room temperature and come back to a soft but solid state.
While the butter is cooling, toast the oats. Preheat oven to 350°F and spread oatmeal on a small baking sheet. Place in the oven for 8-10 minutes. You won’t see a change, but the oats will smell toasted! Remove from the oven and let cool.
Once the butter has solidified, scrape into the bowl of an electric mixer. Cream butter, brown sugar, and white sugar until a little lighter and fluffy in texture. Add egg and beat to incorporate. With the mixer running, add the vanilla extract, baking soda, oats, salt, and flour. Mix to incorporate and finish with butterscotch chips.
Using a cookie scoop, make cookies on a parchment-lined cookie sheet (approximately 1 tablespoon each). Bake in preheated 350°F oven for 10-12 minutes. It is easy to over-bake cookies, so bake until the edges are just set but centers are soft. Remove from oven and leave on the baking sheet for 3-5 minutes to finish baking. Then cool on a wire rack.