Make the beet syrup: Bring 1 cup water, the sugar, honey and vinegar to a simmer in a small saucepan. Simmer until the sugar dissolves, about 10 minutes. Add the beets and simmer until softened, about 20 minutes. Strain through a fine-mesh sieve and let cool.
For each cocktail: Combine 3/4 ounce beet syrup, the lime juice and mezcal in a shaker; shake well. Strain into an ice-filled glass. Add the seltzer and ginger beer. Garnish with a lime wedge.