Adjust an oven rack to the middle position and heat the oven to 350.
Line 3 large baking sheets with parchment paper (I skipped this step, and the cookies didn't stick to the pans).
Whisk the flour, baking powder, baking soda, and salt in a medium bowl.
In a stand mixer fitted with the paddle attachment, beat the butter and sugars at medium-low speed until just combined, about 20 seconds.
Increase the speed to medium and continue to beat until light and fluffy, about 1 minute longer.
Scrape down the bowl with a rubber spatula.
Add the egg and vanilla and beat on medium-low until fully incorporated, about 30 seconds.
Scrape down the bowl again. With the mixer running at low speed, add the flour mixture and mix until just incorporated, and smooth, about 10 seconds.
With the mixer still running at low speed, gradually add the oats and mix until well incorporated, about 20 seconds.
Give the dough a final stir with the rubber spatula to ensure that no packets remain and the ingredients are evenly distributed.
Divide the dough into 24 equal portions, each about 2 tablespoons, then roll between the palms of your hands into balls.
Place the cookies on the prepared baking sheet, spacing them about 2 1/2 inches apart, 8 dough balls per sheet. Using your fingertips, gently press each dough ball into 3/4 inch thickness.
Bake one sheet at a time until the cookies are deep golden brown, the edges are crisp, and the centers yield to slight pressure when pressed, 13 to 16 minutes, rotating the baking sheet halfway through baking.
Transfer the baking sheet to a wire rack and let the cookies cool completely before serving.