Preheat oven to 350 degrees.
Lightly grease 2 baking sheets. In the bowl of a mixer, combine the butter and vanilla. Add 1/2 cup of the sugar, then slowly add the flour and pecans and mix just until combined. Use your hands to roll the dough into balls about 3/4-inch in diameter, and arrange them on the baking sheets. Bake 15 to 20 minutes, until golden brown. Meanwhile, spread the remaining confectioners' sugar out on a plate. When the cookies are done, remove from the oven, let cool 3 to 5 minutes, and then roll each cookie in the sugar until coated, letting each rest in the sugar for a minute to absorb it. Let cool completely on wire racks, then store in airtight containers.
Courtesy Gale Gand, "American Brasserie" by Gale Gand, Rick Tramonto, Julia Moskin, MacMillian, Publishers, 1997