Preheat the oven to 350°F. Prepare baking sheets by lining them with parchment paper or coating with nonstick spray.
In a bowl, sift together the all-purpose flour, baking soda, and salt.
In a separate mixing bowl, cream the softened butter with the granulated sugar until light and fluffy.
Blend in the lemon juice and lemon zest until incorporated.
Gradually fold in the dry ingredients, mixing just until combined.
Stir in the old-fashioned oats, beating until the dough achieves a uniform consistency.
Using a small cookie scoop, portion dough onto the prepared baking trays, spacing evenly.
Fill a shallow dish with granulated sugar and another with water. Press a drinking glass into the water, then into the sugar, and use it to gently flatten each dough portion.
Bake the cookies for 10 minutes, then transfer to a wire rack to cool.
Meanwhile, prepare the glaze by whisking together powdered sugar, lemon juice, and lemon zest in a small bowl until smooth.
Once the cookies have fully cooled, dip the tops into the glaze. Allow the glaze to set before serving.