Preheat oven to 350 degrees. Coat baking sheets with cooking spray; line with parchment, and spray parchment.
With an electric mixer on medium speed, beat butter and sugar until crumbly. Add eggs, vanilla, zest, and salt. Beat until pale and smooth, about 1 minute. Mix in baking powder and oats. Let stand 10 minutes.
Drop batter by the teaspoon, 3 inches apart, onto prepared sheets. Press each into a round with the back of a spoon. Bake until golden, about 9 minutes. Transfer cookies (still on parchment) to wire racks to cool completely. Carefully remove from parchment with a spatula. Cookies can be stored up to 2 days at room temperature in airtight containers.