In a small saucepan, combine the brown sugar and water. Heat gently, stirring just until the sugar fully dissolves. Do not boil. Remove from heat and let cool completely.
Rinse the mint leaves and gently clap them between your hands to release aroma. Do not chop or muddle aggressively — bitterness is not welcome here.
In a large pitcher, combine the fresh lime juice, brown sugar syrup, verjus, mint leaves, and a small pinch of fine sea salt. Stir gently and let sit for 5 minutes to allow the mint aroma to bloom.
Fill the pitcher generously with ice. Top with ice-cold club soda and give one very gentle stir only — just enough to combine without flattening the drink.
Pour into ice-filled glasses and garnish with fresh mint sprigs and lime wedges.