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Brown the butter: Melt the butter and vanilla bean and scrapings (if using) together in a small, heavy-bottomed saucepan over medium heat. (If using vanilla extract or paste, just brown the butter by itself and add the vanilla along with the egg.) Continue to cook, swirling occasionally, until the butter turns golden and smells absolutely amazing, 3-5 minutes. There should be dark brown bits (not black) on the bottom of the pan. When the butter starts to foam up, watch it very closely as it can go from brown to burnt in moments.
Place the sugars in a large bowl and when the butter has browned, scrape it and the browned bits into the sugar immediately to stop the cooking. Let cool, stirring occasionally, for 10-20 minutes. Remove the vanilla bean if using (you can rinse and dry it and use it to make vanilla extract or to flavor a small jar of sugar or liqueur).
Meanwhile, sift together the flours, baking soda, salt and optional xanthan gum into a medium bowl to unclump the oat flour.
Whisk the egg into the cooled sugar mixture until well-combined and emulsified. Use a sturdy wooden spoon to stir the flour mixture into the sugar mixture, stir until well combined, then continue to stir vigorously for 45 seconds; the mixture will firm up slightly. This increases the stickiness of the flours and xanthan gum (if using) helping the cookies hold together and be more chewy. Stir in the nuts and chocolate until evenly distributed.
Cover the dough and let it sit at room temperature for 1-2 hours to allow the butter to firm and the starches to absorb moisture, leading to thicker, chewier cookies. Alternatively, if baking the cookies later, scoop the dough into individual balls, place close together on a small baking sheet lined with parchment paper, cover tightly, and store in the fridge or freezer until ready to bake.
Position racks in the upper and lower thirds of the oven and preheat to 375º. Line two rimless cookie sheets with parchment paper.
Scoop the dough into 1 1/2" diameter balls (about 3 tablespoons each; a size 24 or 30 spring-loaded ice cream scoop works wonderfully) and place on the prepared cookie sheets, spaced 2-3 inches apart. Flick a few flecks of flaky salt over the top of each cookie.
Bake the cookies about 7-10 minutes, rotating back to front and top to bottom after 5 minutes. When the cookies are ready, they will seem underbaked. The edges should be just starting to color, and the tops should be puffed all over with soft centers that collapse when gently touched with a fingertip (note: there will be less collapsing if you didn't use xanthan gum). The centers will look wet under a thin surface of dry, cracked-looking dough.
Remove the cookies from the oven and pull them, parchment and all, onto cooling racks. Let cool for at least 10 minutes before devouring. Or cool completely and store in an airtight container at room temperature; the butterscotch flavor comes through better when the cookies are cool.
The cookies are best within 2 days of being baked.