Preheat the oven to 350F.
In a small saucepan over medium heat, brown the butter. The butter will melt, then bubble, then foam, and then brown. Stir frequently to keep the butter from burning.
Transfer the butter to a bowl and put it in the refrigerator or freezer to cool slightly. You want the butter to be about room temperature and starting to solidify.
Once the butter is cooled, add it to a large bowl and beat the brown butter with the granulated sugar and brown sugar. Beat until the butter and sugar are thickened and combined, about 1-2 minutes.
On medium speed, add the vanilla and the eggs one at a time. Scrape down the sides of the bowl and combine until lightened in color.
On low speed, add the flour, baking soda, and salt. Stir until combined.
Add the oats on low speed until fully combined.
With a wooden spoon or rubber spatula, stir in the butterscotch chips.
Scoop 2 tbsp. sized round balls of dough and then flatten into discs. Place on a parchment lined baking sheet.
Top each with a sprinkle of flaky sea salt.
Bake for about 12 minutes, until the cookies are golden and the edges are firm. You want these cooked through more than say a chocolate chip cookie so they are more chewy than soft.