Heat water in a small saucepan over medium heat. Stir in brown sugar until dissolved. Remove from heat and allow to briefly cool.
For Espresso Martini:
While the brown sugar syrup cools, carefully place a coupe glass into the freezer. It will be chilled after about 5 minutes.
Combine espresso, vodka, coffee liqueur, brown sugar syrup, and egg white in a cocktail shaker. Secure the lid and dry shake (NO ice) for about 20 seconds until frothy.
Add ice, secure the lid, and shake for another 20 seconds until very chilled.
Strain into a chilled coupe glass and garnish with three espresso beans. Serve immediately.
Notes
*I recommend brewing espresso at home OR buying freshly brewed espresso from your favorite coffee shop. You can keep espresso in the refrigerator for about ten hours, so try to purchase it and use it within the same day. If you don't have access to espresso, an excellent substitution is cold brew concentrate.**My favorite coffee liqueur is Forthave Spirits "Brown". Other popular options are Kahlua or Mr. Black Coffee Liqueur. Any should work!***If you'd prefer a vegan version of this Espresso Martini, use aquafaba in place of the egg white. Aquafaba is the liquid that comes in a can of chickpeas, and is a great substitute that still creates a frothy top layer on cocktails.