Make simple syrup: Combine sugar and 1 cup water in a small saucepan over medium heat. Stir until sugar completely dissolves (about 3-5 minutes), then remove from heat and allow to cool completely.
Juice the lemons: While the syrup cools, squeeze the lemons to extract approximately 1 cup of fresh juice. Strain to remove any seeds or pulp if desired.
Blend the berries: Add strawberries or raspberries to your blender with ¼ cup of water and pulse until smooth. For a smoother texture, strain the berry puree through a fine-mesh sieve to remove seeds.
Combine the base: In a large freezer-safe container, mix together the cooled simple syrup, fresh lemon juice, berry puree, and remaining 1¾ cups cold water. Stir well to combine.
Freeze the mixture: Cover the container and place in the freezer for about 4 hours, or until the mixture is partially frozen with a slushy consistency. For best results, stir once or twice during freezing to break up any large ice crystals.
Prepare for serving: When ready to serve, remove from freezer and let stand for 5-10 minutes to slightly soften. Using a fork or ice cream scoop, scrape the mixture to create a fluffy, slushy texture.
Serve and garnish: Scoop into chilled glasses, add a splash of vodka or sparkling water if desired, and garnish with fresh mint leaves and lemon slices for a beautiful presentation.