Burford Brown eggs have become a household name, but what of the hens who lay them?

As well as running The River Café, chef Ruthie Rogers hosts Ruthie's Table 4, a podcast where A-list celebrities discuss food. During one popular episode, Sex and the City star Sarah Jessica Parker offered up a confession: she daydreams about smuggling a particular item back to America when she visits Britain: Burford Brown eggs. ‘They have these orange yolks that I just… oh, my God!' she enthused. She isn't alone: speaking to Kathy Burke on the comedian's podcast, The Traitors presenter Claudia Winkleman revealed that a Burford Brown omelette with mild cheddar would be her last supper.

With their thick, hard, glossy brown shells (which help them to stay fresh for longer) and rich flavour, Burford Brown eggs are beloved of home cooks and chefs, as well as Hollywood stars. Their cachet is such that they are even named on the menus at Claridge's.

The hens which lay them are no less special. My own Burford Browns, which live with me near the Cotswold market town from which they take their name, plod about issuing commanding clucks until they hear the metal feed bin opening, at which point they rush towards it in a flurry of jet black. Generally placid and sociable, each hen has a large, rosy comb and a variable neckerchief of strikingly burnished orange feathers, known as hackles. This is a robust breed suited to gardens made up of tough-stemmed roses and gooseberry bushes or, preferably, farmyards. My dad, upon receiving a pair as a present from me, returned them within the month, declaring that it was as if I had released two rotavators into the garden.

Burford Browns will lay like clockwork from spring into autumn. They can produce up to 240 brown eggs in their first year and, given a good life, can lay for many years to come. When eggs of this hue were first seen, some people believed that they had actually been painted because their colour, with scrubbing, can be removed from the shell.

The hue is unique to specific breeds of hen: the internal shell gland deposits the natural brown pigments, known as porphyrins, at the specific Burford Brown concentration onto the eggshell just before laying. In the wake of bird flu, which has seen poultry shows devoid of live birds to judge, egg showing is increasingly on the rise: a clutch has to be freshly laid, be identical in size and be of its correct breed colour. Unsurprisingly, competition when it comes to the showing of dark-brown eggs has always been especially high.

The Burford Browns' original home was Manor Farm on Westhall Hill, near Burford's main street in Oxfordshire. Back in the 1900s, Mabel Pearman began breeding hens there and noticed that they laid darker than average brown eggs. Their exact genetic make-up remains a closely guarded secret, but their heritage likely includes other traditional breeds known for laying brown eggs, such as Dutch Welsummers and French Marans. Years later, in 1980, Pearman's grandson, Philip Lee-Woolf, would begin to rear the birds himself, remembering the delicious eggs of his childhood. He spent years perfecting his flocks to ensure hardy and uniform genetic traits before taking his eggs, packaged in colourful, characterful boxes, to market under the brand name Clarence Court.

‘The industry was stuck in a rut and dominated by a handful of large packers intent on making profits at the expense of poultry welfare and, in many cases, the producer farmers, too,' he recalls. ‘Quality and individuality didn't seem to matter, and chickens themselves even less. It was all about uniformity, numbers and price — small, medium or large eggs of pale shells and a watery texture, in a drab, grey-pulp box.'

Department stores such as Fortnum & Mason placed orders and then supermarkets. Yet with success came the realities of hard commercialism, as they demanded more and more product, necessitating a larger business model. Philip and his wife, Janet, decided to part with Clarence Court in 2004, knowing that their ethos of having relaxed and properly free-range hens had left an impression on the egg industry and, crucially, consumers.

Philip remains the only person rearing genuine Burford Brown hens for domestic sales; if you have your heart set on owning some, you have to be on his Legbars of Broadway waiting list for his seasonal debutante pullets. If you do get to the top of it, you'll be in good company: the birds reside in coops at esteemed gardens from Gresgarth Hall, Lancashire, to Rockcliffe, Gloucestershire. Jeremy Clarkson's Diddly Squat Farm in Oxfordshire is also home to a flock.

The late Deborah, Dowager Duchess of Devonshire adored her Burford Browns, which she kept at Chatsworth, Derbyshire, and, later, at the Old Vicarage in nearby Edensor. Speaking in 2010 to Jenni Murray on the BBC's Woman's Hour, she proclaimed their eggs as ‘the most delicious… I have ever had'.

They were too good not to share: she arranged for them to be taken up the road to her beloved farm shop and piled into a bowl like a mound of delicate Ferrero Rocher, with empty egg boxes that tempted customers to fill them. Unsurprisingly, they sold out.

The Tithe Barn at Thyme, Lechlade, Gloucestershire, is hosting an exhibition of hen paintings by Arthur Parkinson until May 12.