To make the lavender sugar. Add the lavender to a food processor and pulse until finely chopped. Add the sugar and lemon zest, pulse to combine. In a large mixing bowl, cream together the butter, 1/2 cup lavender sugar, and the vanilla until light and fluffy, about 3-5 minutes. Add the flour and salt. Beat until combined, and the dough begins to form a ball. If the dough is crumbly, add 1-2 tablespoons milk or water until the dough comes together.Roll out the dough between 2 sheets of lightly floured parchment to 1/4 inch thickness. Make sure you're using enough flour or your dough will stick. Cut out the cookies using a 3 inch circle cookie cutter or cut into similar size squares. Carefully transfer the cookies to a parchment-lined baking sheet. I recommend using a spatula to lift the cookies. Cover the baking sheet with plastic wrap and place in the freezer. Freeze until firm, about 10 minutes. Roll out the leftover scraps, and repeat with the dough.Preheat the oven to 325 degrees. Brush each cookie with beaten egg white, then sprinkle with the remaining lavender sugar. If using edible flowers, gently push them into each cookie to adhere. Bake the cookies on the middle rack of the oven for 12-14 minutes or until just lightly golden on the edges. Cool on the baking sheet for 5 minutes, then transfer to a wire cooling rack to cool completely. Cookies can be stored in an airtight container for up to 4 days.