Combine the sugar and 1 cup water in a heavy-bottomed pot. Heat over high heat, stirring occasionally, until the sugar is completely dissolved, 5 to 7 minutes. Remove from the heat and let cool completely.
Pour 1/3 to 1/2 cup of the cooled simple syrup into a 2-quart pitcher. (Reserve the remaining syrup for another use.) Add the lemon juice and 4 cups water and stir to combine.
To serve, fill an 8-ounce glass with ice. Add 2 ounces of the peach brandy, then fill with the peach lemonade to just short of an inch below the rim.
Garnish with a lemon slice and a mint sprig.