To make the lilac sugar, add the sugar and the lilac blossoms to the bowl of a food processor and pulse until combined. The sugar should take on a pale lilac hue. This makes about 1 cup of sugar, in order to fit the bowl of a standard food processor, but you will only need 1/2 cup for the shortbread recipe. (you can make just 1/2 cup of sugar in a small capacity food processor, if you have one)
For the shortbread cookies
Preheat oven to 350F.
Add the flour, butter, 1/2 cup of the lilac sugar, salt, and vanilla extract to the bowl of a food processor. Pulse to combine, continuing until the dough comes together into a single clump.
Remove the dough from the food processor to a lightly floured surface, and roll out the dough into a rectangle. Scatter about half of the whole lilac blossoms over the dough, and then fold the dough a couple times to work them in, and re-roll it out into a rough rectangle. I like to divide my dough in half and do this twice.
Scatter the remaining lilac blossoms over the dough, and roll the dough out until it is about 1/4-1/3 inch thick, making sure the lilac blossoms are well embedded into the dough. Use your fingers to gently press them in, if needed.
Using a 2 inch biscuit cutter, or other circular cutter, cut out about 24 cookies and place them on a baking sheet.
Bake the cookies for about 15 minutes or until just barely starting to turn golden on the edges. Remove and allow to cool on the baking sheet for a few minutes before moving.