Process basil and sugar in a blender or food processor until basil is minced.
Melt butter in a large saucepan; add basil mixture, flour, salt, and cinnamon. Stir in chopped pecans and vanilla; remove from heat.
Drop dough by 1/4 cupfuls onto ungreased baking sheets; flatten to 1/4-inch thickness with bottom of a large glass.
Bake cookies at 300° for 40 minutes or until golden. Remove to wire racks to cool.
*Regular basil may be substituted; increase ground cinnamon to 1 teaspoon.