What difference does it make to use a roux or a beurre manié ?
Are there any rules about their usage, or are they both interchangeable?
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There's a great overview of the differences here, including a taste-test experiment at the end.
Broadly: beurre manié started off as a "lazy" roux; some people claim that the cooking of the roux reduces any "floury" taste; the experiment did not find any discernible difference between the two options for either a bechamel sauce or a velouté.
I like Vicky's answer, but... there's nothing lazy about Buerre manié. It takes hard work to make. It's just faster, like a last minute resort. To make buerre manié, you have to mix the flour into the cold butter.
To make a roux, you have to melt the butter, incorporate the flour, and cook the proteins out of it.
If you're short on roux, you can add buerre manié rapidly, but making a roux is the preferred method.
Another thickener is brown flour. You cook the flour, without butter, until it's light brown. Then you can use it without adding butter.