To make the simple syrup: muddle the blackberries in the water and heat until boiling. Simmer for 5-10 minutes or until the mixture is a deep purple. Add the cup of sugar and bring to a boil again. Remove from heat immediately, strain through a sieve, and set aside to cool.
Juice the lemons until you have one cup of lemon juice. If desired, you can remove the pulp by straining through a sieve.
Mix the lemon juice and water into the pitcher.
Pour in the sweet and condensed milk and whisk until well combined.
Add the ice, then pour the cooled blackberry simple syrup over the top.
Serve immediately, stirring before serving. If desired, garnish with extra blackberries, lemon slices, and mint.