In a large bowl add the melted brown butter, dark brown sugar, and white sugar. Beat together for 1 minute.
Add the egg, milk, and vanilla extract. Cream together until smooth.
Add the dry ingredients: gluten-free flour, baking soda, arrowroot powder, and salt. While scraping down the bowl, mix until combined.
Pour in the oats, and on low speed, mix until fully incorporated. Fold in the butterscotch morsels.
Cover the bowl and refrigerate the dough for 1 hour. (If you chill the dough longer than 1 hour, allow the dough to rest at room temperature for 20-30 minutes prior to baking.)
Preheat the oven to 350 F°. Line 2 baking sheets with parchment paper. Scoop out 1.5 tablespoon-size round balls of dough (35 grams). Roll the dough into a ball, and place the balls of dough onto the tray, about 2 inches apart from one another (about 6 cookies per sheet). Bake for 10 minutes.
Remove the cookies from the oven. Let them cool on the pan for 2-3 minutes. Use a large cookie cutter or glass to round out the cookies while they are warm. Add additional chips and sea salt if desired. Let them cool on the pan for 2-3 minutes, then transfer to a wire rack.
Store the cookies in a tightly sealed container at room temperature for up to 2 days.