In the bowl of a stand mixer fitted with the paddle attachment, combine the butter and powdered sugar. Cream them together thoroughly until fluffy. Add the lemon zest, lemon juice, ground lavender and vanilla and let them get completely incorporated. Turn the speed to low and slowly add the flour just until it disappears. It's important not to overwork the dough at this point.
Turn the shortbread dough out onto plastic wrap. Gently work it together to smooth it out and form it into a thin, long rectangle by gently pressing it together with clean hands. Wrap the dough in plastic and allow it to sit in the refrigerator to chill overnight. This is also going to allow the flavor to really develop.
The next day, pre-heat the oven to 325 degrees and line a baking sheet with a silicone mat or parchment paper. Take the dough out and slice it with a sharp knife into roughly 1 1/2 inch slices to make them pretty little squares. The dough rectangle should be long enough to yield about a dozen shortbread cookies.
Transfer the cookies to the prepared baking sheet and pierce each one twice with a fork for the traditional decoration. Then let them bake for about 18-20 minutes. The cookies should be fully baked but still pale. Take them out and allow them to cool for a couple of minutes on the sheet, then transfer them to a rack to finish cooling.
While the cookies are still warm, dust them with powdered sugar. Then serve immediately! These are so amazing with a great cup of tea.