Combine the strawberries, lemon juice, lime juice, sugar, and 1 cup of water in a blender. Puree until smooth, then add more water a splash at a time until it reaches your preferred taste and consistency.

Fill glasses with ice and pour the frozen strawberry lemonade over the ice.

In a medium saucepan, stir together the sugar and 2 cups of cold water. Bring to a boil over medium-high heat, stirring often, until the sugar fully dissolves, about 5 to 7 minutes. If desired, add a few fresh mint leaves and let them steep as the syrup cools, then remove the leaves. Let the syrup cool completely.

Add the strawberries to a blender and process until completely smooth. Pour the puree into a large pitcher and set aside.

Pour the lemon juice and lime juice into the pitcher with the strawberry puree. Add the cooled simple syrup to taste and stir well. If the lemonade is too strong or too sweet, adjust with cold water until the flavor is balanced.

Chill in the refrigerator or serve right away over ice cubes.