We have gotten to where we go to Europe every other year around Thanksgiving. We go every other year because we build up mileage and it seems it takes about that time to earn the mileage for our family to go in Business Class.
We've pretty much gone everywhere.
We absolutely love this time frame for a few reasons:
Weather is great (nice and cool, sometimes cold but not terrible)
Plus depending on when Advent is, the Christmas Markets open up around much of Europe and are amazing. We especially enjoyed Germany, France and Hungary for the markets. Just really a lot of fun and kind of magical.
Way less crowds at this time of year all over most of Europe. Especially since obviously Thanksgiving isn't a holiday they celebrate

. However, we've seen Thanksgiving dinners offered in many of the major cities. I have always skipped these, because we'll Thanksgiving Dinner wouldn't be right if not done in 'Merica!
So for us, Late November is perfect. Prices are also better. You will run into some things not being open depending on where you are. We ran into this a few times when we were in the Czech Republic.
And the snow falling in the UK or German or Austria or Czech Republic or Hungary is pretty amazingly pretty.
One more thing I'll add in reference to May or September. In September in France, for example, much of the local stores will close for "holiday", it seems like it used to be August this happened, but lately it's creeping much more into September and much of Europe can still be quite warm depending on where you go. I've been to Paris in September when it was pretty hot and not everything has A/C like the USA.
frankly, it's just easier to pack for a longer trip when the weather is colder in my opinion. You can layer much easier and not worry about getting into crazy heat. Light Wool clothing is great to layer and is also very hygienic