Has anyone tried this and did it work? I'm curious if it would give a similar, or maybe even better result compared to hand whisking.
Edit: I don't have a milk frother right now, and was thinking of buying one for this.
I've never used anything other than a spatula for any 'roux'-type sauce.
I'd think any hand-held milk frother would either be far too delicate or simply under-powered. The first round of milk addition to a roux/béchamel mix is almost a solid. It never gets down to milk thickness even when fully finished. I really think it would kill something as low-powered as a milk frother. It would grind to a halt [battery] or burn out [mains]. I wouldn't even dream of using a full-powered stick blender on a roux.
Answering for some additional information:
Technically, you could use a milk frother for making bechamel, in the same sense that you could also use a pair of chopsticks or a large fork (I've done both, while travelling). The question is whether it would be a good tool for making it. And the answer is no.
In addition to the underpowering issues that Tetsujin mentions, there are the issues that most frothers:
Lemme explain the last, because it's why you don't really want a powered appliance for making bechamel at all. Generally, you whisk bechamel relatively slowly (you do the same with polenta, custard, etc.). You don't want to whip air into it. Most powered mixers of any kind (frothers, handheld mixers, stick blenders) are designed so that their lowest speed is still way too fast, and will result in incorporating air into the bechamel, which you don't want. It might even prevent the bechamel from thickening properly.
One thing worth noting is that such frothers (at least from my experiments) struggle with thicker liquids. I have one, and have enjoyed frothing milk up. Then I thought: well surely it's going to be perfect for whipping cream.
But it's not (at least not in my tests). I'm guessing it's either too localized or weak to allow much air to be incorporated in a thicker liquid. It would always froth the cream up a bit, but then start to diminish in effectiveness, quickly getting slowing down (despite full batteries), and never really reaching the mass whipping cream gets when you use a mixer, and it seems even inferior to regular hand-whisking it.
I see the base recipe for béchamel only may call for milk... so that aspect may be ok unless opting for a richer sauce base. But still, by being thickened, I really wonder if it might not have the power to perform well. In thinner liquids, a frother's quick vibrations in a small space allow enough turbulence to even incorporate air, but I think with thicker liquids, its small scale or weak strength quickly would limit its usability.