Place the sesame seeds in a large, dry skillet over medium-low heat. Toast for approximately 4–6 minutes, stirring continuously, until fragrant and lightly golden. Remove from the heat before the seeds become dark brown.
Immediately spread the toasted seeds on a large plate. Let cool for approximately 10 minutes. Do not leave them in the hot skillet, where they can continue browning.
Place the cooled seeds and 2 cups of the measured cold water in the blender. Blend for 1–2 minutes, or until the seeds are finely ground and the liquid looks milky. Follow your blender’s operating limits and pause if necessary.
Set a fine-mesh strainer lined with a clean nut-milk bag or fine straining cloth over a large bowl. Pour in the blended mixture. Let it drain, then gently squeeze the bag or cloth with clean hands to extract the liquid.
Return the strained sesame pulp to the blender. Add the remaining 2 cups measured water and blend again for approximately 30–60 seconds.
Strain the second mixture into the same bowl, gently squeezing to extract the liquid. Set aside the remaining solids; they are not part of the finished drink.
Add the sugar to the strained sesame liquid. Stir until completely dissolved.
Transfer to a measuring pitcher. Let excess foam settle briefly, then add enough cold water to bring the finished drink to exactly 4 cups. Stir thoroughly.
Taste and adjust the sweetness if needed, adding sugar 1 teaspoon at a time and stirring until dissolved.
Cover and refrigerate for approximately 1 hour, or until thoroughly cold.
Stir well before serving. Divide among four glasses, approximately 1 cup per serving before ice. Add optional ice to individual glasses.