Combine the lemons, limes and oranges in a medium bowl and toss with 1/4 cup of the sugar to coat. Sprinkle with a generous pinch of salt.
Place the fruit, cut-side down, on the grill; cook until dark brown grill marks appear, 5 to 8 minutes. Remove to a plate and let cool. Cut 2 lemon halves into slices and set aside for garnish.
Combine the remaining 1 1/2 cups sugar with 2 cups water in a saucepot over medium heat. Cook until the sugar is fully melted, about 5 minutes. Let cool completely.
Pour the sugar syrup into a pitcher and add the basil leaves. Juice the grilled citrus through a strainer into the pitcher. Add vodka to taste, if using. Stir, add ice, and garnish with the reserved lemon slices. Serve cold.