Combine the water and sugar in a small saucepan, warm just enough for sugar to dissolve. Add to the lemon juice and refrigerate for 1 hour or until chilled. Freeze the lemonade in an ice cream machine according to manufacturer's directions until it reaches a slushy consistency. Add the candied lemon peel. Serve with a straw and a spoon.
Bring 4 cups sugar and the water to a boil in a 1-quart saucepan. Stir until the sugar dissolves and remove from the heat. Set aside.
Place the lemon zest in a separate saucepan and cover with cold water. Bring to a boil, drain the zest, and rinse under cold running water. Repeat this process 2 more times, using fresh water each time.
Place the blanched zest in a saucepan and cover with enough of the sugar-water syrup to cover by 1-inch. Bring the syrup to a boil and reduce heat to a simmer. Cook the zest until it is tender and translucent in color, about 10 minutes. Remove from the heat, cool, and drain the zest. Lay zest on a sheet pan to cool, then toss in the sugar until well coated. Transfer to an airtight container and store at room temperature for up to 1 month.