Begin making the fresh peach puree by placing the peach slices, sugar and water in a small sauce pot. Bring to a boil then reduce to a simmer. Let cook down for about 20 minutes or until softened. Use a potato masher to gently puree the peaches; set aside to cool. Place in refrigerator until you're ready to use.
To make the mules, place the mint sprigs and lime juice in the bottom of a large pitcher. Using a muddler or the back of a wooden spoon, begin gradually mashing the mint sprigs to release the flavor and natural oils. Add the peach puree, gin, and ginger beer to the pitcher and stir well.
Fill four copper mule mugs with ice and fill 2/3 the way up with the peach and gin mixture. Top each glass with club soda and garnish with lime wheels and fresh mint to serve.