Preheat the oven to 375 degrees F. Line two baking sheets with parchment paper.
Grind the oats in a coffee grinder or food processor and place them in a bowl. To the oats, add flour, baking powder and salt and stir well. Set aside.
In a mixing bowl, beat butter until creamy. Beat in the sugar and lemon zest and continue beating until light and fluffy. Beat in the vanilla extract and the egg.
Stir the flour mixture into the sugar mixture until blended, then stir in the lemon juice to make a soft dough. It should not be too sticky.
Spoon the dough up by tablespoons and roll into balls. Dip the balls in yellow sanding sugar, crystal sugar or granulated sugar. If you're planning on icing the cookies, you can skip this step (or use sugar AND icing like I did).
Arrange about 2 inches apart on parchment lined baking sheets and press down lightly with bottom of a glass.
Bake the cookies one sheet at a time on center rack for about 10 minutes or until edges are browned. Let cool on baking sheets for 3 minutes, then transfer to a wire rack to cool completely.
To make the lemon icing, melt the butter in a 2 cup microwave-safe liquid measuring cup. Add the sugar to the cup and stir it around with the butter, then add lemon juice 1 teaspoon at a time and stir until the icing is a drizzling consistency. Drizzle over the cooled cookies.