When straining things that clog up the screen, I tend to use a spoon or spatula to move the stuff around, so you can get expose un-obstructed parts of the screen.
Of course, you might end up pushing some of the pulp through the process, but it's generally minimal.
You can then either dump out the pulp as you go (possibly into muslin or cheese cloth, let it cool, then give that part a squeeze), or use the back of a large spoon or a rubber scraper to push the pulp against the strainer to get out some extra liquid. (or use an oven mitt, covered in a plastic bag ... it's messy, but it works).
Of course, if you have a cheap strainer, too much force on it can cause the whole screen to pop out, splashing you and making a quite large mess. (I've since made sure that my strainers weren't basically a screen just held in by pressure to a loop of metal that attached to the handle, but are actually one integral part).
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If you're looking at buying new equipment, I'd look into getting a chinoise with the proper sized stick (not sure what it's called), which you can use to basically compress all of the pulp into the bottom as you're working.
If the soy is "sweetened", I would not suggest using it in a savory sauce. However, if it is unsweetened, it should be ok. I personally like soy, but it all depends on the brand, and whether full fat, sweetened, plain or vanilla as to how I would use it. You might try baking with it, perhaps muffins, cake or biscuits - and substituting the soy for whatever liquid you would otherwise use.
Best Answer
Ok - given your clarification try the following:
This is exactly how I make mine (I make both varieties though), and there is very little (if any) beany taste left. In fact, sometimes as a result (mind you I use the same beans everytime) it comes a little bit nutty.