Drain the soaked soybeans and boil them in fresh water for 20 minutes, until a bean squeezes apart easily between two fingers.
Drain again, then blend the beans with 1 L / 4 cups of fresh cold water, the sesame seeds and pine nuts until completely smooth — a full 2 minutes in a strong blender, longer in a weaker one.
Strain through a fine sieve or a layer of cheesecloth for a silkier soup, or skip this if a little texture doesn't bother you.
Chill the soy milk at least 2 hours, or speed things up with a handful of ice cubes stirred in just before serving.
Cook the noodles in boiling water 3 to 4 minutes, until tender, then rinse immediately under cold running water while rubbing the strands between your hands to strip off the surface starch.
Divide the noodles between two bowls and pour the chilled soy milk over them.
Top with cucumber, tomato and the egg, and drop in a few ice cubes to keep everything cold at the table.
Serve salt and sugar on the side and let everyone season their own bowl — this is genuinely how Korean families still disagree about it.