Preheat the oven to 325 degrees F (165 degrees C). Cover baking sheets with foil, then coat the foil with a nonstick cooking spray.
Combine oats, flour, salt, and baking powder in a medium bowl. Beat sugar and butter in a large bowl until fluffy. Beat in egg and vanilla. Mix in flour and oat mixture until just combined. Drop dough by teaspoonfuls onto the prepared baking sheets, 2 ½ inches apart.
Bake cookies in the preheated oven until edges turn golden, about 10 to 12 minutes. Cool cookies on baking sheet, about 10 to 15 minutes, then peel them off with your fingers. Repeat with remaining dough, re-spraying baking sheets with nonstick cooking spray between batches.