Most countries in the EU have a VATOSS/VATMOSS filing system. They're all relatively straightforward, but Ireleand's system is generally the easiest to use...and also the only one in English. (This matters because it means that they'll correspond with you in English, and any tax authorities in other EU countries are on notice to correspond with the taxpayer in English.)

A few blogs mentioned that as an external US entity, you probably want to do VAT with Ireland (basically you deal only with Ireland for paying VAT, and what you pay gets forwarded through the system to all the other EU countries).

The reasons they cited were that it's in English, they have among the shortest initial application times (a few weeks vs. a few months for others), and that they're generally the most business friendly (which you can guess given how they attracted all the US tech companies as as tax haven).

In practice I've found this to be the case. I've had a few email chains with the Ireland tax authority, and they were responsive within a few days. And I got things done. Though I was surprised to find that the "best" EU tax authority was still worse than the IRS.

Things I was surprised by: 1) It takes an extremely long and complicated application form (and several weeks) to get an EU VAT number to pay VAT taxes. By comparison the IRS lets you get an EIN instantly, and you only need to fill out a few fields 2) Setting up a login to the Ireland Tax system is the worst thing in the world. I expected a simple username/password type system. Instead: you apply, then you have to remember a code, they don't give you any confirmation email, then 3 weeks later they send you an email giving you another code, and you have to hold onto that code, because you need to wait 72 hours for the database to update, before you can enter that code to create a certificate (but you don't know to use the first code or the second code), and you then need to save that certificate to your local machine, and re-upload it whenever you want to login. It's actually even worse than I describe, but I honestly cannot even remember all the steps 3) When filling in how much VAT you owe each country, you need to fill out every country and every amount manually, no way to upload a CSV or anything

If this is how bad Ireland is, I imagine other EU countries are even worse.