A day ahead, make the peach nectar. Bring a medium pot of water to a boil over high heat. Fill a medium bowl with water and add the ice cubes. With a sharp knife, lightly score an "X" at the bottom of each peach. In 2 batches, put the peaches in the boiling water and cook for 1 minute. Submerge the cooked peaches in the iced water for 1 minute. Pat dry and peel. If needed, use a vegetable peeler. Cut the peaches into large slices and remove the pits. Place the peaches in a medium nonreactive bowl, and toss with lemon juice.
Add the peaches and juice to a blender. Add ¾ cup water. Blend until pretty smooth. Pour the peach juice into fine mesh strainer over a large bowl. Use a large spoon to scrape the strainer multiple times, so the juice can pass through. Discard the pulp of the peaches.
Transfer the juice to a medium sauce pan. Add sugar. Bring to a boil and until the sugar has melted. Remove from heat and cool to room temperature. Once cooled, pour into airtight container and keep in the fridge until ready to use.
To make the cocktail, tear the mint leaves with hands and put into a cocktail shaker. Muddle a few times. Add bourbon whiskey, ginger liqueur, peach nectar and 10 ice cubes. Close the shaker and shake vigorously.
In a glass cup, add 4 - 5 ice cubes. Strain the cocktail into the cup. Add club soda. Give a quick stir. Garnish with peaches and mint sprig. Serve.