Combine the ingredients in a cocktail shaker without ice and shake for 30 seconds until egg is emulsified. Add ice to the shaker and shake for aonther 30 seconds. Strain into a coupe glass and grate fresh nutmeg over top.
Combine sugar, water, and grated ginger in a small saucepan. Bring to a simmer over medium heat, stirring often until sugar is dissolved. SImmer for 10 minutes, reducing heat as needed, to allow ginger to infuse. Remove from heat and let cool to room temperature before using. Syrup will keep in an airtight container in the fridge for up to two weeks.