Milk temperature for white sauce

As you see from the variety of advise from reputable sources, many combinations of hot/cold roux and liquid will work.

From a convenience point of view, you want at least one of them hot in order to speed the integration. If you started both of them cold, it would probably work but take a while to warm up to melt the butter in the roux, and free the flour particles to integrate and create the sauce@mdash;and it will won't fully thicken until the mixture is at a boil, so this is inefficient. You would probably also have to mash and stir more frequently to prevent local burning.

If you start with both hot, it will also work, although you might have to whisk relatively quickly—but who has both a hot base and a hot roux sitting around? Usually, at least one was prepared in advance, even if only by a few minutes. Batali is recommending this method, but note that in the professional kitchen, where he lives, time is the most constrained resource, so this is his method of choice. His line cooks probably have hot roux ready at all times.

A hot roux is easier to scoop and measure.

A hot liquid will quickly melt and dissolve the roux.

But in any case, if the roux is properly made, the starch granules are surrounded by fat, and so are not going to cause lumping in any case, despite Ramsay's advise. As soon as the mixture is warm enough to melt the butter, the particles are going to move away from one another, and won't be able to clump together before they are hydrated.

So I would not worry about this in the slightest.

In my experience the secret to great roux sauce is to add the milk very slowly at the beginning. Keeping the hob temperature low, add a splash of milk to the butter/flour mixture, stir, and repeat. Gradually increase the amount of milk added in each turn, and soon you'll have a smooth white sauce with no lumps. When the sauce is looking more liquid than solid you can increase the hob temperature slightly too, but don't over-do it as the sauce might curdle.

I haven't found any difference between using room-temperature milk, and milk straight from the fridge. The biggest factor is not adding too much milk at the earliest stages - patience is the key!

I have had trouble when adding cold milk to very hot roux. The first milk to hit the pan thickens too fast into a paste that I then have to work out of the sauce. I have had decent luck using a cool roux with cold milk, but the smoothest and fastest sauce has been with hot roux and hot milk, added 1 cup at a time.

If you heat the milk there is no need to add it slowly or in tiny amounts. Heat it in the microwave, and While it's heating, crush together the butter & flour with a fork. Then add the hot milk to the pan and scrape in the butter/flour mix. The hot milk gets to work on the butter/flour and immediately the flour does its job and thickens, it's prevented from lumping by being mixed with the butter; stir briskly until it's all incorporated, then add the cheese and flavourings. You'll thank me!!!