Traditionally, onions are slowly braised in butter before being drained, puréed and added to a pre-prepared Bechamel sauce to make Soubise sauce.
This seems a terrible waste of flavour, if you were to make the Bechamel last of all with the onion-flavoured butter, would this be too overpowering and would it ruin the final sauce?
(I'm assuming here that any water added to the butter during the braising/poaching process would not effect the thickening of the milk/cream and would have evaporated off).