Preheat your oven to 350 degrees F (175 degrees C). Lightly grease an 8"x8" pan.
Take half the oats and grind in a blender or mini-food processor till fairly finely ground. Set aside till later.
Cream the butter and sugar until light and fluffy, then add the oats, ground oats and flour. Beat until everything is incorporated.
Add the milk only if needed to get the dough to stick together. Be careful not to add too much milk if using any at all as the dough should be very dry and crumbly, but should stick together when pressed.
Press into the prepared pan and poke with a fork a few times before putting in the oven to bake for 18-20 minutes until lightly golden and until the top isn't oily looking anymore.
The shortbread will not be completely cooked at this point. Remove from the oven and cool for 5-10 minutes while preparing the lemon filling.
Whisk together all the filling ingredients in a medium-sized bowl. Pour over the already baked oat shortbread and return the pan to the 350-degree F (175-degree C) oven for 18-20 minutes. The edges will brown fairly quickly but don't worry. Once the filling is set and doesn't jiggle anymore it's done. Remove it from the oven and cool before serving with a light dusting of powdered sugar over top if desired.