Chill two martini glasses, Vodka, and dry vermouth in the freezer.
To prepare the clarified watermelon juice, begin by dicing the watermelon into manageable pieces that will fit into the mouth of a blender. Blend. Pour melon puree into a glass pitcher and leave in the refrigerator up to 2 hours. Spoon off mealy froth and discard, then pour the balance of the remaining juice through a fine strainer. Reserve.
While the watermelon puree is separating, prepare the simple syrup. To make the simple syrup, place 1 cup each of water and sugar into a pan and bring to a boil and all sugar has dissolved. Remove from heat and let cool before placing in the refrigerator.
Separate the yolk and white of 1 egg. Draw out 1/3 of the white
When you are ready to mix the cocktail, combine Vodka, watermelon juice, simple syrup, and egg white in a cocktail shaker. Shake rapidly for 1 minute. Remove glass from the freezer, pour in the Vermouth, swirl it around the glass, and then dump out whatever remains. Line the glass with sugar (here you can use superfine sugar if you have it). Add ice to your cocktail shaker (This part is a little messy and can be done in the sink). Shake for another twenty seconds, then strain into prepared glass.