Using a food processor, grind the pecans and 1 tablespoon granulated sugar for 20 seconds; transfer to a medium bowl.
In the food processor, pulse the butter, remaining 1/3 cup granulated sugar and the vanilla until combined, scraping the bowl once or twice. Add the flour and pulse to combine. Stir into the pecan mixture.
Position racks in the upper and lower thirds of the oven and preheat to 325 degrees . Line 2 cookie sheets with parchment paper. Roll 1 tablespoon dough into a ball, press a caramel into the center, then roll to enclose the caramel. Repeat with the remaining dough and caramels; place 2 inches apart on the prepared cookie sheets. Bake until lightly colored, about 15 minutes. Let cool for 15 minutes; coat with confectioners' sugar while warm. Coat again just before serving.