In a medium saucepan, combine the water and sugar. Add the sprigs of mint. Cook on medium heat, just until the sugar is dissolved. Let cool to room temperature. You can speed this along, by placing the sugar syrup in the refrigerator.
Discard the mint and pour the sugar syrup into a blender. Add the lemon juice and strawberries. Purée until smooth and refrigerate for about 1 hour. If you are using frozen berries, you can cut this time to 30 minutes.
To make the sorbet, pour the purée into the bowl of a countertop ice cream maker. Churn until it is firm, about 20 minutes. Place in the freezer to firm up more, about 4 hours.
Meanwhile, prepare the lemon cups by cutting the lemons in half length-wise. Use a spoon to scrape out all of the fruit. Rinse and pat dry.
Fill each lemon cup with about ¼ cup of the sorbet. Place the cups in a freezer-safe container. Freeze again until ready to serve, up to 3 hours. If freezing longer than 3 hours, cover the container to prevent freezer burn.