Whisk together the flour, baking powder and salt in a small bowl.
Beat the sugar and butter with an electric mixer on medium speed in a large bowl until light and fluffy, about 5 minutes. Beat in the eggs and then the vanilla. Add the flour mixture and beat on medium-low speed until completely incorporated. Scoop out heaping tablespoonfuls of the dough, roll with your hands into balls and refrigerate until firm, about 30 minutes.
Position oven racks in the top and bottom thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
Put some sugar in a small bowl. Roll the chilled dough balls in the sugar and arrange on the prepared baking sheets about 2 inches apart.
Bake until the bottoms and edges are just barely golden and the tops are no longer glossy, 14 to 16 minutes, rotating the baking sheets about halfway through. Let the cookies cool on the sheets for a few minutes, then transfer them to a wire rack to cool completely. The cookies can be stored in an airtight container at room temperature for up to 3 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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