Make the ginger purée: In a blender, purée the ginger, sugar and 1⁄4 cup water until smooth.
Make the cocktail: In the bottom of a cocktail shaker, muddle the peach, mint leaves and 3⁄4 ounce of the ginger purée until it has the consistency of a rough purée. Add the simple syrup, lemon juice and bourbon, fill with ice and shake vigorously. Strain into a rocks glass filled with ice, garnish with a peach slice and mint blossoms (if using), and serve.