The white part is a ceramic that has metallic traces plated onto it like a circuit board before the TEC is built. The TEC is then soldered together with the 'couples' sandwiched between the two 'circuit boards'. Ceramic is used because it needs to be reasonably thermally conductive, while being an electrical insulator.
It's quite possible that the ceramic would crack without tearing the metallic traces that were plated on. As long as it stays clamped in place it will probably work fine. It may not last long without careful handling though.
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