The solution is to incorporate in the member state where you live (and perform the services), and invoice your clients for your services as a company.
If you want to hire a French employee, you can incorporate a French subsidiary, and you invoice his wages (+ some operating expenses) from your main company to the subsidiary. Doesn't cost much. Clean and simple.
I have been speaking to a French lawyer about this exact scenario (UK company employing a French resident).
I've been quoted a cost of around 8% of salary to be retained by this subsidiary- hardly insubstantial.
Alternatively they can put a framework in place so that the UK company can directly employ in France. Here the costs are lower but coming in at at least 3000 euros is by no means small change.
I suppose the cheapest option (I haven't investigated) would be for the French employee to incorporate themselves and deal on a corp/corp basis. However the admin cost is then incurred by them and they wouldn't be covered by us on French employment terms.
The admin for employing a UK person is vastly easier and cheaper. We already have the contracts and accounting in place. I have been very surprised that cross-border hiring is so difficult/messy/costly in Europe.
If you want to hire a French employee, you can incorporate a French subsidiary, and you invoice his wages (+ some operating expenses) from your main company to the subsidiary. Doesn't cost much. Clean and simple.
How is this a problem at all?