Hmmm. The title got me thinking, and this came out. Not thought through, but seems credible.
The tempo.
A simple circular counter with a clock face on it (symbolic, not functional)
Each player has one Tempo which replaces their f-pawn. Or black could choose which pawn - same for both.
The tempo can move one square in any direction, just like the king. However, the tempo cannot be captured, or capture another piece. Nor can it castle or be promoted. A tempo blocks the square it occupies and moving their tempo ends a players turn - it is just another piece in all other regards.
In play: if a player moves any of their pieces onto a square adjacent to the opponents tempo, without giving check, they get an extra go. A tempo. They cannot earn another tempo on this free turn, even if they land a piece next to the opponents tempo again. The clock stays running for them until the 2nd move is made - it is mandatory.
Could be radically different ways to approach the play... Maybe.
The board is 100x100
and put a lot of pawn ♟ and general