Place an oven rack in the center of the oven. Preheat the oven to 325 degrees F. Butter and flour an 8 by 8-inch glass baking dish.
For the bars:
In a small bowl combine flour, thyme and salt. Set aside. Using a stand mixer fitted with the paddle attachment, beat together 1 stick of butter and powdered sugar on high speed until light and fluffy, about 30 seconds. Beat in the lemon juice and vanilla. Reduce the speed to low and gradually add the flour mixture. Using damp fingers, press the dough into the prepared pan. Bake for 30 minutes until golden. Cool for 30 minutes.
For the glaze:
In a medium bowl, whisk the lemon juice and powdered sugar together until smooth. Spoon the glaze over the cooled crust. Allow the glaze to harden, at room temperature, for at least 1 hour.
Using a metal spatula, remove the crust from the pan. Cut into 1 1/2-inch square bars and arrange on a serving platter or store airtight in a plastic container at room temperature.