chess - No moves at all, not even to put yourself in check

Here, white is stalemated. The king is not in check, but white has no legal moves. Not because any move white makes would put the king in check -- which is how stalemates normally happen -- but because no white piece can move at all, even if we temporarily ignore the no-self-check rule.

(Black is not stalemated: king takes bishop.)

Now, this example is a bit wasteful, 62 white pieces plus the two kings. Also, it can't be reached from the starting position.

Come up with an example, using the fewest pieces (total both sides). Can it actually be reached from the starting position?

Small hint:

there is essentially only one answer. [EDIT: OP screwed up]

Answer

Here are my first idea (both sides are essentially the same answer, so the hint fits too):

Both positions seem to be independently reachable by a legal game. It might be possible to find a legal game leading to the whole position too, but that would take a bit of time.

Before that, I'm going to double check for any simpler solutions. :-)

Since OP commented that only one side needs to be stalemated in this rigorous fashion, this should do the trick: