Muddle the quartered strawberries with the Ginger Syrup in a cocktail shaker. Add the bourbon and lime juice, fill the shaker with ice and shake well. Strain into a highball glass filled with ice. Top with a splash of soda, garnish with a mint sprig and whole strawberry and serve.
Combine the ginger juice and sugar in a small saucepan and bring to a boil. Cook, stirring, until the sugar dissolves. Remove from the heat and let cool completely. Keep covered in the refrigerator up 2 weeks.