To make the lemon cups: Trim a very thin slice off of the lemon bottoms so they'll sit sturdy when standing up. Cut off the top 1/3 of the lemon and reserve to cap the cups. Carefully hollow out each lemon with a teaspoon, turning and digging into the pith so the entire inside of the fruit comes out, take care to leave the lemon shell in tact. Chop and mash the lemon pulp and strain into a blender. (You should have about 1 cup of fresh lemon juice, reserve it for the filling.) Freeze the lemon cups in egg cartons, with their lids, at least 1 hour or up to overnight. The frozen shell will prevent the lemon ice from melting too quickly when you serve it.
To make the ginger syrup: Combine the sugar, water, and ginger in a pot over medium heat. Gently simmer and stir for 2 minutes until the sugar is dissolved and the ginger fragrant ¿ do not allow the syrup to boil or get dark. Remove from heat, cover, and steep until cool.
Fill a blender with crushed ice, pour in the reserved cup of fresh lemon juice, add the ginger syrup (remove the big pieces of ginger), and process until slushy. Add the lemon zest and blend again. Freeze for 1 hour to firm it up a bit. Scoop the lemon ginger ice into the prepared lemon cups, garnish with mint leaves, top with the lemon cap, and serve. Serve in egg cups for a great presentation.