Combine the sugar and 1 cup water in a small saucepan over medium heat.
Simmer, stirring occasionally, until the sugar fully dissolves and the syrup runs clear, about 3 minutes.
Remove from the heat and let the syrup cool to room temperature, about 15 minutes.
In a large pitcher, combine the lemon juice, cooled syrup, and 3 cups cold water, stirring well.
Taste and adjust: add a little more syrup if it tastes sharp, or a splash more water if it tastes too strong.
Cover and refrigerate the pitcher for at least 30 minutes, until fully chilled.
Right before serving, fill glasses with ice and pour the lemonade base in until each glass is about two-thirds full.
Top each glass with sparkling water, stir once gently, and garnish with a lemon slice and a mint sprig.
Good to Know
Double the lemonade base and keep it covered in the fridge for up to three days: top with sparkling water fresh each time you refill, rather than all at once.