Adjust oven rack to middle position and heat oven to 350 degrees. Line 3 large (18- by 12-inch) baking sheets with parchment paper.
Whisk flour, baking powder, baking soda, and salt in medium bowl.
In standing mixer fitted with paddle attachment, beat butter and sugars at medium-low speed until just combined, about 20 seconds. Increase speed to medium and continue to beat until light and fluffy, about 1 minute longer. Scrape down bowl with rubber spatula.
Add egg and vanilla and beat on medium-low until fully incorporated, about 30 seconds. Scrape down bowl again.
With mixer running at low speed, add flour mixture and mix until just incorporated and smooth, 10 seconds.
With mixer still running on low, gradually add oats and mix until well incorporated, 20 seconds. Give dough final stir with rubber spatula to ensure that no flour pockets remain and ingredients are evenly distributed.
Divide dough into 24 equal portions, each about 2 tablespoons (or use #30 cookie scoop), then roll between palms into balls.
Place cookies on prepared baking sheets, spacing them about 2 1/2 inches apart, 8 dough balls per sheet (see note above). Using fingertips, gently press each dough ball to 3/4-inch thickness. Lightly sprinkle sea salt evenly over flattened dough balls before baking.
Bake 1 sheet at a time until cookies are deep golden brown, edges are crisp, and centers yield to slight pressure when pressed, 13 to 16 minutes, rotating baking sheet halfway through.
Transfer baking sheet to wire rack; cool cookies completely on sheet.