In a food processor with knife blade attached, pulse pecans with 3 tablespoons confectioners' sugar until very finely ground.
In a small bowl, combine flour, baking soda, salt, and ground pecans.
In a large bowl, with mixer on medium speed, beat butter and sugar 1 minute or until creamy, occasionally scraping bowl with rubber spatula. Add egg and vanilla; beat until well mixed. Reduce speed to low; gradually beat in flour and pecan mixture just until blended, occasionally scraping bowl.
Shape dough by rounded measuring teaspoons into 1-inch balls. Place balls, 1 1/2 inches apart, on ungreased large cookie sheet.
Bake cookies 13 to 15 minutes or until bottoms are browned and cookies are light golden. Let stand on cookie sheet 2 minutes to firm up slightly, then transfer to wire rack to cool. Repeat with remaining dough.
Place confectioners' sugar in pie plate. Roll cooled cookies in sugar to coat, twice if needed.
Note: Make sure cookies are completely cooled before rolling in the confectioners' sugar; if they're still warm, the coating will get sticky.
Store cookies, putting waxed paper between layers, in tightly sealed container at room temperature up to 1 week or in freezer up to 3 months.