Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large bowl, sift together the flour, baking powder, baking soda, and salt.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter, brown sugar, and confectioners' sugar on medium-high speed until light and fluffy, about 2 minutes. Add the peanut butter and beat to combine, scraping down the sides of the bowl as needed. With the mixer on low speed, slowly add the oil, vanilla, and egg and beat until just combined, about 20 seconds. Add the flour mixture, beating just to combine.
Lightly flour the palms of your hands, and roll the dough into balls about 1/2 inch in diameter. Place them about 1 inch apart on the prepared baking sheets. Bake, rotating the sheets halfway through, until the cookies are golden brown on the edges, about 12 minutes.
Transfer the baking sheets to wire racks and let cool completely. (The peanut butter cookies can be stored, unfilled, in an airtight container at room temperature for up to 1 week.)