Preheat the oven to 300 F
Combine the lavender buds and 1/4 cup of the sugar in a food processor fitted with the metal blade. Pulse the processor four or five times to combine and then run the machine for about 10 seconds to pulverize the lavender and sugar. (This processing step goes a long way towards dispersing the lavender essential oil into the sugar!)
Put the lavender sugar into the bowl of a stand mixer fitted with the paddle attachment. Add the remaining 1/4 cup sugar and the butter. Beat the sugar and butter on low until the mixture is smooth and creamy, but NOT fluffy. Add the flour and combine well to make a cohesive dough.
Transfer the dough into a nine inch square baking pan or into a Nordic Ware English Shortbread Pan, which also measures 9 inches by 9 inches. Press the dough into the pan, smoothing the edges to get a tidy edge. (Pressing the dough into the Nordic Ware pan ensures that the dough with take on the pan's floral impressions.)
Refrigerate the dough in the pan for about twenty minutes prior to baking to let the dough rest.
Bake the shortbread in the middle of the preheated oven for about 20 to 25 minutes or until the shortbread is just slightly colored. Let the shortbread cool in the pan on a rack for about five minutes and then turn it out onto the rack to continue cooling. Cut the shortbread into 9 pieces when almost cool.
Store the shortbread in an airtight container. It also freezes very well.
Serve with tea, ice cream, and/or fresh seasonal berries!