Combine first 3 ingredients in a small saucepan. Cook over medium-low heat 1 to 2 minutes or until butter melts and sugar dissolves, whisking constantly. Remove from heat. Stir in vanilla.
Combine ground pecans and flour; add to molasses mixture, stirring until blended.
Working in batches, making 2 cookies at a time, drop batter by 1 tablespoonful onto a lightly greased aluminum foil-lined baking sheet (keep remaining batter covered with a damp cloth).
Bake at 350° for 8 minutes or until golden. Let stand 2 minutes (no longer). Working quickly, carefully roll each delicate cookie into a tight cigarette shape. Cool completely on wire racks. Serve within 3 hours.