In a medium bowl, whisk together the flour, baking powder and salt. In a separate large bowl using an electric mixer, beat together the butter, granulated sugar and brown sugar, mixing until pale yellow and fluffy, 3 minutes. Add in the vanilla and eggs and mix well. Slowly incorporate the dry ingredients into the wet ingredients and mix well.
Using slightly wet hands, roll the dough into heaping tablespoon-size balls (or use a small cookie scoop) and place them about 2 inches apart on the prepared baking sheets. Bake just until the edges start to brown, 10 to 12 minutes. (Bake one sheet at a time for even cooking and browning.) Transfer the cookies to a wire rack and let cool.
The cookies will keep in an airtight container for up to 3 days.