Inn to Inn hiking/walking recommendations with delicious cuisine Europe?

The first place that comes to mind that is my personal favorite in Europe is the English Lake District. There are many towns and villages in and amongst the hills and mountains, farms along the way, plenty of spectacular scenery and lots of places to stay from farms, to B&Bs, youth hostels, and hotels. You'll find lots of fresh, locally grown food and see plenty of farms in the area that raise mutton, beef, and other stock. Not so much agriculture here but plenty of real English rural beauty! There are lots of companies that will arrange it for you but also very easily done on your own - we've hiked there extensively and keep returning because we enjoy the area so much.
We have also done hut-to-hut hiking in the alps, particularly in the Dolomites. It isn't so much village-to-village but the mountain refugios you stay in have excellent food and you'll meet interesting people from all over.
I'm sure you'll get lots of other suggestions over the next few days but these are a couple ideas to start with!

I too enjoyed walking northern England and the Lake District but "delicious" will depend on one's pallet. I enjoyed the full English breakfasts that fueled a days walk with only a light lunch to break the day's walk. Evening meals were pub food but apparently pub food has come a long was from the old joke about hell being English cooks.

A number of booking agencies will organize and book accommodations for self guided walking holidays. We found Contours Walking Holidays highly satisfactory. You could of course do it yourself. http://www.contours.co.uk/index.php

You may want to contact Rob and Emma at Mountainbug Holidays in the French Pyrenees (nice couple, well-organized, beautiful hiking) or William at High Point Holidays (based in France but also has England self-guided trips). Have fun!

There are many choices! I read about this trail, so cannot offer personal experience but it sounds amazing - enjoy planning and have a great trip!

http://www.via-alpina.org

I ran into a German couple who were doing that in Sudtirol, Italy last month (Sept). They were going from Brixen to Merano . There are many trails and the scenery is excellent. Food and wine is wunderbar along the way. You can also buy a Mobilcard which gives you unlimited access to the regional buses and trains if you feel like taking a break from the hiking. We spent 7 days in the area and it was fantastic.
http://www.sii.bz.it/en
http://www.suedtirol.info/en/

Here are a couple of options to get you started:

The Wicklow Way (Ireland)
Other Irish marked walking routes
The Alpine Pass Route (Switzerland)
European Ramblers Routes

In Germany, you can hike the Rheingau. There are all types of accommodations, it's the region known for Riesling, and there is good cuisine to be found. I was shocked to get unbelievably delicious salmon at the Jagdschloss Niederwald in Assmannshausen/Rüdesheim. The hikes will give you expansive views of the Rhein and its castles. Hikes are in the moderate range.

I see you run a Farm to School programme at home so I would recommend walking in Lazio above Rome. We came across a wonderful eco friendly farm which is run by as a vegetarian eco-friendly agriturismo. The couple who run it are wonderful cooks and their farm has astonishing Roman remains which Stefano is delighted to show guests.
It is in a good walking area and although we were on a "self-guided" tour with an English travel company (On Foot Holidays) there is plenty of scope for independent walking. Most of the other places we stayed on that trip were far from vegetarian but all very strong on local and seasonal food.
http://www.lemolesulfarfa.com/what-to-do/