First whisk the flour, baking powder, baking soda, lavender and salt together in a bowl to aerate the dry ingredients. Set it aside. Then set up a stand mixer with the paddle attachment and cream the sugar and butter together until it is a fluffy base. Pour in the lemon zest, lemon juice and eggs next and let them get completely mixed in. Turn the speed to low and slowly pour the dry ingredients in until it becomes a soft dough. Line a sheet tray with a silicone mat and lightly flour it. Turn the dough out onto the tray and pat it out into an even, 1/2 inch thick layer of dough that almost fills the sheet with well floured hands. Stick the tray in the freezer for 20-30 minutes to make it completely firm and easy to work with.
When time is up, get out another sheet pan and line it with another silicone mat. Lightly flour it too. Then use a donut cutter to cut as many donuts and their holes out of the layer of dough as possible. Lay them out on the second sheet tray, then put the tray back in the freezer to completely firm up the donuts for another 20-30 minutes. While that is happening, get out a large cast iron pot and fill it with about 3-5 inches of canola oil. Use a deep fry thermometer to make sure it gets to and stays at around 350. Then quickly make the glaze. Combine the powdered sugar and juice of two lemons in a large bowl and stir it together thoroughly until it is a thin glaze.
At this point the donuts and holes should be ready. Take them out and fry them in batches so that the pot is not over crowded. It will only take about a minute on each side for them to become gorgeously golden and puffy. When the first batch is done, remove them to a plate lined with paper towel to drain and get the next batch in. While the next batch cooks, dip the finished hot donuts and holes in the glaze and transfer them to a rack set over a lined tray. Sprinkle them with the lavender vanilla sugar on top before the glaze sets. Repeat this process until all of the donuts are cooked, glazed and sprinkled with sugar. Let them cool completely, then store them in air tight containers for up to a day!