Crush the lavender and set aside.
In a mixing bowl, beat the butter and 1/2 cup sugar for 2 full minutes.
In a separate bowl, sift the flour, add the lavender and stir to combine.
Add the dry ingredients to the butter mixture and mix until crumbly. Add 1 tablespoon water and mix until a dough forms. If it is too crumbly, add another tablespoon of water. If too sticky, add a tablespoon of flour.
Roll the dough into a log on parchment paper. Wrap in plastic wrap and chill for at least 1 hour.
Add 2 drops of purple food coloring to the remaining 1/2 cup sugar on a sheet of plastic wrap. Roll the chilled dough log in the sugar. If the sugar does not stick, brush the log lightly with egg white and roll again.
Slice into cookies about 1/8 to 1/4 inch thick and place on a cookie sheet 2 inches apart. Bake at 350 degrees for 8 to 10 minutes, or until lightly golden.