Preheat oven to 325 F (160 C) and line two large rimmed baking sheets with parchment paper or silicone baking mats.
In a medium bowl stir together the sugar, orange zest and lime zest to make a citrus sugar mixture using a fork or small whisk and let sit a couple of minutes.
Transfer the citrus sugar to the bowl of a stand mixer fitted with the paddle attachment. Add the softened unsalted butter and vanilla, then beat together on medium for 3-5 minutes until light and fluffy.
Add the flour mixture to the creamed butter and mix again on medium low until a cookie dough consistency is formed.
Dough should be moist but firm. (if you feel it is dry add a tablespoon of milk.dough should NOT be sticky)
Remove the dough from the bowl and divide into two halves. Shape each half into a disc and wrap in plastic wrap.
Place the dough in the fridge to chill for 30 minutes.
Place one half on a gently floured surface and roll out to ¼ inch thickness.
Use a 2.5-3 inch cookie cutter to cut out as many cookies as you can.
Transfer cookies to prepared baking trays, spacing 1-2 inches apart.
Using a pastry brush, gently coat a thin layer of beaten egg white onto the center of each cookie. Then, gently press a flower blossom slightly onto the surface. The back of the blossom may need to sink into the dough for petals to lay flat. Sprinkle lightly with sugar.
Place the entire tray into the fridge for 10-15 minutes to make sure cookie dough is cold before baking.
Place trays on the center rack of the oven. Bake for 14-16 minutes or until edges start to slightly golden in color. Bake time may differ between ovens, so keep an eye on the first batch.
Place cookie sheet on a wire rack and let cool for 5 minutes. Then, using a spatula, transfer cookies off the pan and directly onto the wire rack to cool completely.
Repeat with remaining half of dough.