Heat the oven to 375°F with racks in the upper- and lower-middle positions. Line 2 rimmed baking sheets with kitchen parchment. In a 10-inch skillet over medium, cook the butter, swirling the pan occasionally, until the milk solids at the bottom are deeply browned, 4 to 6 minutes. Pour into a large bowl and set aside.

Distribute the oats in an even layer on one of the prepared baking sheets; distribute the pecans in an even layer on the second baking sheet. Place the oats on the lower rack and pecans on the upper rack. Toast until the pecans are fragrant, about 2 minutes; remove from the oven and set aside to cool. Stir the oats, then toast until light golden brown, another 3 to 5 minutes. Transfer the oats to a food processor and cool until barely warm. Reserve the parchment-lined baking sheets.

Meanwhile, whisk both sugars into the browned butter. Whisk in the whole egg plus egg yolk and the vanilla until the mixture is aerated and pale in color.

To the cooled oats in the processor, add the flour, salt, baking powder and baking soda. Pulse until the oats are finely ground, 7 to 10 pulses. Add the oat-flour mixture to the egg mixture; fold with a silicone spatula until well combined and no dry patches remain. Add both chocolates and the toasted pecans (reserve the parchment-lined baking sheet); fold until evenly distributed. Cover and refrigerate for at least 30 minutes or up to 1 hour.

Using a ¼-cup dry measuring cup or a 2-inch spring-loaded scoop, drop six mounds of dough, evenly spaced, onto each reserved baking sheet. Bake until golden brown, 12 to 15 minutes, rotating the baking sheets and switching their positions about halfway through.

Set the baking sheets on wire racks. Sprinkle each cookie with flaky salt (if using). Cool for 5 minutes. Using a wide metal spatula, transfer the cookies directly to the racks. Cool completely before serving.