In a large mixing bowl using an electric hand mixer, or in the bowl of a stand mixer, add the softened butter and ⅔ cup of granulated sugar. Cream until light and fluffy. Add the lemon zest and vanilla and stir to combine.
Let the cookies cool on the tray for 5 minutes, then transfer to a wire rack to cool completely before adding the glaze. To make the lemon glaze, mix together the lemon juice, powdered sugar and vanilla.
Glaze the cookies with the icing by dipping a fork into the glaze and drizzling it over the cookies while they are on the rack. Any excess will drip off. Give the glaze 15 minutes to set before serving or storing.