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Re: European master's as stepping stone to Top 10 US PhD
It would be more meaningful if you could post your current profile, as not everyone needs a full-fledged masters to be competitive. If you're looking at LSE's EME, then definitely. The median placement is around Top 20, while those with Distinctions in their MSc get into Top 10. However, do realise that it's an insanely difficult programme.
Notwithstanding, UK masters programmes (especially the one listed above) generally have little to no opportunities for RA work, to the best of my knowledge.
You can get letters, but you shouldn't apply in December of this year. That's because none of your masters grades will be in, by the time you apply. So there's no signaling value, apart from the fact that you enrolled in that programme.