In a cocktail shaker, combine the ginger syrup, lime juice, vermouth, and amaro. Strain the cocktail into the highball glass.
Fill a highball glass with 1¼-inch cubes. Fill the cocktail shaker with ice, cover, and shake vigorously.
Add the 2 dashes of Angostura bitters to the top of the cocktail and top with the club soda. Skewer the orange half-wheel through the ends and place the mint sprigs through the center, then garnish the cocktail with the orange mint flag.