Bring the sugar, 1/2 cup water and ginger to a simmer in a small saucepan. Tear and add in 4 of the mint leaves. Once the sugar is dissolved, about 2 minutes, turn of the heat. Strain and set aside to cool.
Slice the peach into slices, reserving 2 for garnish. Add the peaches, the remaining mint, and the lime quarters to a cocktail shaker. Mash with a muddler until broken up and juicy. Add the bourbon and muddle again. Add 2 ounces of the simple syrup, fill the shaker with ice and shake.
Strain into 2 cups filled with ice. Top off with ginger beer, garnish each with a peach slice, a lime slice and a sprig of mint.