Center the oven rack; preheat the oven to 350F (180C). Line 2 (or possibly 3) baking sheets with parchment paper.
Tip the flour, baking powder, baking soda, and salt into a medium-size bowl. Whisk thoroughly to combine.
In the bowl of a standing mixer outfitted with the paddle attachment (or in any large bowl if using hand-held electric beaters), beat the butter and the white and brown sugars at low speed until just combined. Boost the speed to medium , and beat until the mixture turns pale and fluffy -- about 5 minutes. Scrape down the bowl if necessary. Beat in the egg and vanilla.
At low speed, beat in the flour mixture. Stir in the oats.
Use a cookie scoop that holds 2 tablespoons to form the cookies, spacing them about 3 inches apart on the baking sheets. Use your hand to gently flatten the cookies.
Bake the cookies until they color lightly -- about 16 minutes, depending on your oven.