In the bowl of a stand mixer fitted with a paddle attachment, beat together butter and sugars until pale and fluffy, 3 to 5 minutes. Stop the mixer, scrape down the bowl and add flour, salt, and lavender, Beat on low speed until dough just comes together. Stop the mixer, remove the bowl, and incorporate the rest of the ingredients with a spatula.
Dump the crust mixture into the prepared pan, and use your fingertips to press the dough into the bottom of the pan. Bake for 18 to 20 minutes, until slightly browned.
To make the filling: while the crust is baking, in a medium bowl, whisk together eggs and sugar until light and pale. Add flour, lemon juice, and lemon zest. Whisk until completely blended and incorporated.
Cool completely in the pan. Cut into wedges and dust with sifted powdered sugar and dried lavender buds. To store lemon bars, place in an airtight container in layers, separated by was paper. Lavender lemon bars will last, in an airtight container, in the refrigerator, for up to 4 days.