In a large skillet, place the quartered strawberries, sprinkle on 1/2 cup granulated sugar, and allow to sit for 5 minutes to begin releasing the juices. Add the 4 cups of water and bring the mixture to a boil over high heat. Reduce the heat to a simmer and cook the strawberries for about 20 minutes to release all of the flavor. Taste the juice; if it is not sweet enough, add a tablespoon of sugar at a time until it reaches your desired sweetness. Remove from heat and allow to cool until lukewarm.

In a food processor or blender, process the strawberry mixture until smooth. Strain through a fine mesh strainer and discard the solids. Add the lemon juice and refrigerate until cold before serving. If the strawberry mixture is too thick, thin it with a bit of water—you won't sacrifice the flavor. Garnish with strawberry slices and a sprig of mint.