Stir together the sugars and lavender. You can do this ahead of time so it really flavors the sugar!
Place the cold butter in a stand mixer and beat on medium speed. Beat in the brown sugar and granulated sugar and beat on medium speed until combined, about 3 to 4 minutes. Beat in the vanilla extract until combined.
Beat in the flour until it is just combined. If it is still very floury, you can shake in a few drops of vanilla extract to bring it together! Beat in the white chocolate.
Remove the dough from the bowl and separate it into two pieces. Roll each piece in a log that is about 2 inches thick. Wrap it tightly in plastic wrap and refrigerate for at least two hours.
Preheat the oven to 350 degrees F. Place about ½ cup coarse sugar on a plate.
After two hours, unroll the log. Brush it with the egg wash. Roll it through the coarse sugar so it sticks.
Slice the log into ½-inch slices. Slice down gently and if the white chocolate chunks break the dough apart, you can push the pieces back together. You want rounds on the baking sheet about 2 inches apart.
Bake the cookies for 10 to 12 minutes, until they are slightly golden. Remove and sprinkle with a pinch of flaked salt. Let cool completely then serve!