Add peach chunks, lemon slice, vodka, and peach liqueur to a cocktail shaker. Muddle to combine.
Stir in peach nectar. Pour over a glass full of crushed ice. Top with club soda and garnish with peach slice, lemon slice, and sprig of fresh mint.
Notes
If you can't find Mathilde peach liqueur, you can substitute peach schnapps. It's not my preferred ingredient, as it has that artificial peach flavor, but it will do in a pinch.
Fresh, ripe summer peaches work best for this recipe. They give the cocktail an authentic peach flavor.