Place half (4 ounces/115g) of the (cold) cubed butter in a bowl of a stand mixer and put the other half in a medium skillet or saucepan. Melt the butter over medium heat, stirring and scraping the bottom and sides of the pan frequently with a heatproof spatula. The butter will start to sputter and foam, and then begin to brown. Keep cooking the butter just until you see browned bits floating it in. Remove from the heat and pour it over the cubed butter in the mixing bowl, scraping any and all brown bits in the pan into the butter. Let cool for about 30 minutes, until the two butters start to become solid again.
While the butter is cooling, put the flour, salt, and baking soda in a food processor. Pulse it once or twice then add 1 cup (100g) of the oats and half (~1 cup/150g) of the chopped pecan brittle. Pulse until the ingredients are finely ground together.
Attached the paddle to the stand mixer and add the brown and granulated sugar to the bowl and mix for two minutes at medium speed but don't overwhip the mixture (which can cause the cookies to spread too much) until thoroughly combined and smooth.
Scrape down the sides of the bowl with a spatula then add the eggs and vanilla and mix 1 minute more, then stir in the flour/oat/brittle mixture and continue to mix until there are no dry spots. You may want to stop the mixer and do this final step by hand so the ingredients get combined properly. Add the remaining pecan brittle bits and oats and stir until they are well-combined with the dough.
Scoop the dough into 1/4 cup (2oz) rounds on a parchment covered small baking sheet. (You can also make them half that size. See headnote.) Cover the dough tightly and refrigerate for 12 hours, or up to 4 or 5 days.
To bake the cookies, preheat the oven to 350ºF (180ºC). Place the oven racks on the bottom and top third of the oven and line two baking sheets with parchment paper. Place 6 rounds of dough evenly spaced apart (by at least 3-inches/7cm) on the baking sheets and bake the cookies, rotating the baking sheets midway during baking, turning them from front to back, and switching the baking sheets from the top to bottom racks in the oven, until they are light golden brown across the top and deep golden brown around the edges, about 16 minutes. (In Claire's book, she says 16 to 20 minutes, so best to judge them by their color rather than rely on precise minutes.)
Let the cookies cool for 5 minutes, then transfer them to a wire cooling rack. Bake the rest of the cookies on the baking sheet, and it's okay if it's still slightly warm. Just be aware the cookies baked on a warm-ish baking sheet may take a little less time to bake.