Place strawberries in a bowl and add 3 tablespoons sugar. Roughly mash them with a fork until the sugar and the strawberries are incorporated. Cover and let sit for 30 minutes at room temperature.
Blend the strawberries until you get a puree. Then strain through a fine sieve so catch all the seeds onto a bowl. Set aside.
Boil 2 cups water and add 1 cup of granulated sugar. Stir to combine until completely dissolved. Set aside.
To a large pitcher, add the remaining 5 cups of cold water, the sugar-water mixture, the strawberry puree (strained), and freshly squeezed lemon juice. Stir to combine. If not sweet enough, dissolve more sugar in hot water and add to the pitcher until desired sweetness. Note: if making ahead, let it sit in the fridge overnight at this point.
When ready to serve, add ice, sliced lemons, sliced strawberries and mint leaves for garnish. Serve cold.