It was a turnip.

“…As you can see, the ladies are using this kind of, er, round radish,” Ehab, the guide, was saying. I glanced at the tour group around me – Americans, Swedes, Germans. They were all nodding.

I let Ehab finish, then took him to one side.

“It’s a turnip,” I said.

“I think some kind of white beetroot,” he said. “It grows in the ground.”

You can’t blame him. In the Middle East I’ve only ever seen turnips served sliced and pickled as a rarely-sampled table garnish. But here, in a vaulted Ottoman kitchen in the Palestinian city of Nablus, half a dozen local women were unmistakably preparing whole stuffed turnips for lunch.

As we prepared to start peeling and coring, Ehab opened his iPad, asked me to type “T-U-R-N-I-P” into Wikipedia, then sat in the corner scrolling and tapping. Meanwhile, I tweeted a Lebanese chef friend of mine in London, Anissa Helou, who’s written widely on Middle East food. She’d also never heard of stuffed turnips.

“Palestinian cuisine is very interesting,” she tweeted back.

It is. And, for obvious reasons to do with conflict and political isolation, it remains mostly unknown outside the towns and villages from where it originates. You could count on one hand the number of companies taking tourists into the West Bank – and they all have political axes to grind.

Breaking Bread Journeys, a new Palestinian and Israeli joint initiative, is trying to be different. Its purpose is explicitly non-political. The founders – Elisa Moed, who is Israeli, and Christina Samara, who is Palestinian – both already run their own mainstream tour companies. They met in 2010 during discussions on tourism development brokered by Tony Blair’s Quartet peace talks. Realising that they shared perspectives, the two women decided to work together.

They drew up a series of itineraries that – uniquely – cross back and forth between Israeli and Palestinian destinations, linking sites and experiences normally shut off from each other. Their project has just launched with a food tour. Hence the turnips.

Having trotted through Nablus souk buying spices and sampling deep-fried pastries in syrup (“Healthy food!” grinned the vendor), we’d ended up at Beit Al Karama, the House of Dignity – a local women’s project. Its director, Fatima Qadumy, explained that besides running classes in maths and English and supporting women’s empowerment, Beit Al Karama is also the sole Palestinian link to the worldwide Slow Food movement, preserving and documenting the region’s culinary traditions.

That said, there was nothing slow about the way these ladies were coring turnips, deftly stuffing each with a mixture of rice and meat seasoned with cinnamon. Flash-fried to seal their outsides, the turnips then stewed in a thick sesame broth before being turned out into Armenian-style blue-and-white serving dishes. Their peppery-sweet tang, set off by the gooily aromatic rice stuffing, was lifted deliciously by lemon-scented lentil soup, crisp rocket salad and fresh-baked flat bread. We were light on leftovers.

All next day was spent roaming the alleyways of Jerusalem’s Old City in the company of the Chefs For Peace, a group of Christian, Jewish and Muslim chefs who volunteer their time to promote cross-cultural cooking. As we shouldered a path through Jerusalem’s markets, sampling sweet sesame halva and pausing for a plate of hummus, the group’s founder Kevork Alemian explained how a distrust of politicians sparked the idea to lead by example.

“We are simple people,” he said. “When we unite in a kitchen, we forget about our differences.”

The evening peaked at Bulghourji, a splendid Armenian restaurant tucked into Jerusalem’s 16th-century walls, discussing techniques and recipes in the kitchen, then sitting together for a heavenly spread of Levantine comfort food, pastries of melting goat cheese jostling for attention with delicately spiced ground lamb kubbeh.

Crucially, Breaking Bread also showcases cooking where it matters: in the home. Among glass-fronted bookcases and prints of domed skylines in her Beit Safafa sitting room, Iman Abuseir told us her family’s story of exile in the 1930s and 40s from their home – now in Israel – before serving us soup of green wheat, stuffed cauliflower leaves and the classic Palestinian dish musakhan, chicken flavoured with pine-nuts and the lemony spice sumac.

Then in her West Jerusalem apartment of sideboards and willow prints, Ahuva Guterman – an ultra-Orthodox Jewish great-grandmother – made a blessing over dough as she encouraged us to knead our own plaited loaves of challah, a sweet white bread served for the ritual Friday night meal. Wreathed in baking smells, Ahuva then smilingly told us of her parents’ experience of anti-Semitism in 1930s’ Poland and their self-imposed exile to Palestine.

As throughout, we were left to draw our own conclusions. Personally, I would have preferred a bit more politics on the tour. Without context, there’s always a danger you suggest that quality of life and opportunities for Palestinians and Israelis are equal, which they’re not.

But food and ideology rarely mix well, and this is definitely a trip for pragmatists, making linkages between cultures that neither side likes to acknowledge. It leaves you feeling that the physical barriers restricting travel in the Middle East are insignificant, compared to the mental ones.