On the menu are fluffy pittas packed with chopped salad and meats coated in flavour-packed spices; colourful salads; hummus pots topped with zhoug (a Yemenite chilli paste that at Sesame is laden with fresh coriander and ground cardamom) and chickpeas; pots of thick Greek yoghurt sprinkled with toasted black and white sesame seeds and drizzled with date syrup.
Ironically, the long journey to market has meant they have opened at precisely the right time. Although the cuisine is already well-established at the top end of the foodie market, in restaurants like the Palomar and their original venture, Ottolenghi, only now is it really taking hold in the fast-food sector.
"It's exploding," says Ronen Givon, one of the founding partners of Hummus Bros - a similar operation, specialising (surprise, surprise) in hummus, falafel and a menu drawn from the Levant - or Eastern Mediterranean countries, which include Israel, Syria, Lebanon and Turkey.
Givon and business partner Christian Mouysset have been serving mezze-style meals based on their hummus and Israeli-style pittas, with toppings such as mushrooms, mashed fava beans, sun-dried tomatoes and feta and various side salads, for the past 10 years.
Their business has grown comfortably - expanding to four stores in that time - but Givon has recently noticed a sea change in attitude.
"Until a year ago I had to convince people this bowl of Middle Eastern food would be a legitimate lunch," he laughs.
"But lately I feel the graph has tilted upwards. We were serving 200-300 customers a day at our Cheapside store only a few months ago, and it has now jumped to between 400 and 700."
"The flavours are the same but suddenly people are warming up to them."
Uri Binay - owner of East London-based chain Pilpel - came here from Israel 16 years ago, and opened his first store in 2009. Pilpel's menu focuses on falafel, with hummus and a range of salad accompaniments plus the same, proper, fat pittas his competitors offer.
He has also seen a growing interest in his falafel business. "When I first started, no one knew what falafel was, but now I get buyers from all the major supermarkets coming in to take pictures and taste what I'm doing. I think it's the freshness and exotic nature of the food that makes it so popular. Recently I've been getting offers to open worldwide from places like Dubai and Australia - I've now had hundreds of emails."
Sesame's Bar has also noticed the move towards this type of food.
"People are always looking for the next best thing - it's a bit of a fashion. Years ago it was all about French food and cream sauces. Then Italy and the River Café style of food was all the rage. Now it's food from the near East, which is a great cuisine - healthy with most of the fat from vegetables. We are lucky to have been born in the Med and can bring what we eat here."
Bar does not believe that the spread of Med cuisine is simply a trickle-down effect from the more high-profile Middle Eastern/Med-influenced restaurants.
"Ottolenghi food is a different cuisine than this street food. Our influence has been more on chefs and dinner party cooks. The Israeli fast-food restaurants in London are a trend that would have happened here without Ottolenghi. Ronen (of Hummus Bros) and Uri (of Pilpel) would have done their stuff anyway."
According to Givon, Israeli street food has also entered other sectors.
"We have been contacted by large catering companies to do pop-ups in many of the big corporate canteens. We now serve hummus and falafel in companies like Google, JP Morgan, Goldman Sachs and a load of other offices based at Canary Wharf or in the City as well as at Chelsea FC or Wembley Stadium. Their employees need to stay in the building, so we go to them. We can serve 500 customers on the day we are in one of these canteens."
Givon feels it's a coming of age and likens the change in attitude to the coffee revolution.
"There's a maturity to the market - and flavours like za'atar and harira are becoming familiar."
And just in case we were in any danger of becoming used to these powerfully savoury tastes, Sesame also offers new flavours like amba (a tangy mango pickle) and zhoug, and will be introducing new dishes like gondi - Persian Jewish meatballs - picked from a range of cuisines. This variety of influences is something, Bar explains, that he and his design team have tried to reflect in Sesame's colourful interior design.
Bar and Givon are both clear that the foods they are offering are not just Israeli, but a mash-up of the best of what is on offer from the Mediterranean - from the Atlantic to Greece via Cyprus, Palestine, Lebanon and North Africa.
