Preheat oven to 350° F and line baking sheets with parchment paper.
Add sugar and lilac flowers to a food processor and blend until flowers are completely broken down, 2-3 minutes. Scrape vanilla bean seeds into food processor and blend again until mixed.
Pour this lilac-vanilla sugar into the bowl of a stand mixer fitted with the paddle attachment. Add butter; beat until blended, about 1 minute.
Add eggs, one at a time, scraping down the bottom and sides of the bowl in between. Beat until blended, about 1 minute.
Add vanilla and salt; mix.
Add flour, cream of tartar, and baking soda and beat on medium until thoroughly blended, about a minute or two.
Place lilac sugar in a small bowl. Roll cookie dough into balls, about 1 ½ tablespoons each. Roll cookie dough balls in lilac sugar until coated on all sides.
Place cookies onto prepared baking trays, about two inches apart. Bake until lightly golden around the edges, about 9-11 minutes.
Allow cookies to cool for a few minutes before moving to a cooling rack.