Muddle the kumquats, mint, and honey in a cocktail shaker. Add the vodka and ice and shake, shake, shake until chilled.
Add the kumquat slices and mint sprigs for garnish into two champagne or coupe glasses. Pour the cocktail into the glasses, dividing it evenly. Top each glass with ¼ cup of ginger beer. Serve chilled.
Notes
I picked coupe glasses to serve this cocktail, which look like small bowl-like champagne glasses. Champagne glasses would also be perfect, as would small martini glasses. This cocktail is so pretty; it will look gorgeous in whatever glass you choose!