Mix simple syrup, lemon juice, 1/4 cup peaches, and about 1/2 cup ice in a cocktail shaker or plastic cup. Using the end of a wooden spoon, muddle mixture until peaches start to break up. Pour in vodka, cover shaker with top (or the cup with plastic wrap), and shake until blended.
Divide unstrained mixture between 2 tall glasses. Add half the remaining chopped peaches to each, then fill glasses with ice. Top each glass with 1/4 cup club soda and garnish with a peach slice.
*Find next to the cocktail mixes at a well-stocked beverage store.