Make the ginger simple syrup: Bring water and sugar to a boil and stir until sugar dissolves. Remove from heat and stir in the ginger. Allow to steep for at least 30 minutes. Strain into an airtight container. Store in the refrigerator.
Place apple butter, vodka, ginger syrup and apple cider in a cocktail shaker or glass jar with a lid. Shake vigorously.
Fill a rocks glass with ice. Pour the cocktail into the glass. Top off with ginger beer. Garnish with rosemary and cinnamon stick.