There should be a single white piece. We need to put the king in check (requiring a white piece) but we want to make it as easy as possible to get back out again (so any other white pieces are either superfluous or actively detrimental).
We're trying to maximize how many ways there are to break the check. That means moving the king, getting between the king and the offending piece, or eliminating the offending piece. A trivial consideration will determine that the white piece should not be a knight or a pawn, as they don't allow the range necessary for truly large numbers of blocks.
There's no reason to include any black pieces other than knights, queens, and the one king. We're trying to maximize the number of black options, and all other pieces have strictly fewer options than a queen.
The king's ability to move can be safely ignored. Any space that he might move to other than directly away from the threat could instead be filled by a unit that would have at least one blocking move, and directly way from the threat will not save him anyway.
Further, the king and the threat should be as far away from one another as possible, as this maximizes the number of squares that one could interrupt
Having the attack be a straight rather than a diagonal is preferred. Both allow the same number of squares to interrupt (either by blocking or by killing the target) but the straight allows more pieces on either side of both the king and the threat.
and thus...
The optimal configuration is a broad avenue. Black King at e1, white rook at e8, columns d and f filled entirely with black queens, and columns c and g filled entirely with black knights. The knights on rows 1, 2, and 8 have 1 unchecking move each (6x1), as do the queens on 1 (2x1). The queens on 2 and 8 have two unchecking moves each (4x2), as do all other knights (10x2). All other queens have three unchecking moves (10x3). Total unchecking moves is 66
