Watch the butter as it browns. It can go quickly from almost-browned to perfectly browned. Keeping the butter in motion with the whisk is key.
Melt the butter in a heavy-bottomed skillet set over medium heat, whisking constantly, then cook until lightly browned bits form in the bottom of the skillet and the butter is light tan in color with a nutty aroma, about 8 minutes. Remove from heat; butter will continue to brown even after removed from heat. Transfer butter to a medium bowl and place in the refrigerator. Chill for at least 1 hour.
Adjust oven racks to upper and lower third of the oven and preheat oven to 350°. Line two baking sheet pans with parchment paper.
In a medium bowl, whisk together the oats, flour, baking powder, salt and cinnamon. Set aside.
To the bowl of a stand mixer fitted with the paddle attachment, add the browned butter, dark brown sugar and sugar (Alternately, use a large bowl and a hand-held electric-mixer.) Beat on high until the mixture is fluffy, about 3 minutes. Scrape down sides of the bowl with a spatula. Beat in the eggs and vanilla until just combined. With mixer on low, add the oat mixture in 2 batches, beating just until combined.
Drop 2-inch balls of dough onto prepared baking sheets using a tablespoon or small ice-cream scoop. Flatten slightly using the bottom of a glass. Bake, rotating and baking sheets midway through baking, until cookies are uniformly light brown, 12 to 14 minutes. Let cookies cool on baking sheets until set, about 2 minutes, then transfer to a wire rack and let cool completely. Serve.
The dough can be refrigerated for up to three days or frozen for up to two months.