Khan was born in July 1969^[1][2] and grew up in Calcutta.^[3] She has an older sister^[4] and a younger brother. Her family lamented the birth of a second daughter instead of the desired son.^[4] According to Khan "there is a deafening silence" in India about the disappointment a family feels at having a second daughter.^[4] She has said that she and her siblings were treated equally by her parents, and that she and her mother "made peace while I was still very young."^[5] Her father is Rajput from western Uttar Pradesh.^[3] Her mother is from West Bengal and had a catering business in the 1970s and 1980s.^[3][6] According to Khan, her father and grandfather worked to unionize laborers in India.^[7] Khan attended La Martiniere in Calcutta.

Khan had an arranged marriage and immediately afterward moved with her husband to Cambridge in 1991.^[8] She had never learned to cook^[9] and missed the dishes she had grown up eating.^[10] She first started learning to cook from an aunt who lived in Cambridge.^[11] After her aunt died, Khan returned to India for a visit of a few months^[10] to continue lessons with her mother and the family's cook.^[11][12] She told Francis Lam that learning to cook from her mother helped their relationship.^[6]

In 1996 her husband moved to SOAS University of London to teach,^[11] and Khan started studying law at King's College London. She graduated with a PhD in British Constitutional Law in 2012.^[3]

In 2012,^[13] after obtaining her PhD, Khan started offering a series of private supper clubs in her home for a dozen people for £35 each.^{[3][11][14][7]} Vivek Singh attended one and invited her to host a pop-up at his restaurant, The Cinnamon Club.^[10] Eventually the supper clubs grew to serving 45,^[7] and in 2015, after her family complained,^[14][15] she moved the supper clubs to a Soho pub, Sun and 13 Cantons, and started serving lunches instead.^[13][11][16] She struggled early on to lead her team of home cooks, mostly local immigrants working for the first time in a professional kitchen, but then a review by Fay Maschler helped bring in customers.^[13][14][15]

Simon Quayle, a regular customer and executive director of Shaftesbury, offered her the opportunity to compete for a lease in Soho.^[10] When an investor backed out, Khan's husband gave her the money she needed, although he disapproved of the business.^[3] Khan opened Darjeeling Express, a 56-seat location^[7] offering Indian Rajput and Bengali^[13] home cooking specialties,^[4] in 2017 in Soho.^[3] She named the restaurant after a train she rode as a child during summer holidays.^[17] Her team consisted of Asian immigrant women who all had full-time jobs and were never trained professionally; they worked for Khan on their days off but were able to cut back hours at their other jobs and eventually give them up altogether.^[14][18] Most of the all-female kitchen staff^[8][19] at the restaurant are second daughters.^[4] All are South Asian.^[1] Food & Wine called Darjeeling Express a "smash hit".^[20] Khan was profiled by the BBC in a short documentary.^[10]

In 2018 her cookbook, Asma's Indian Kitchen,^[3] was published by Pavilion Books.^[21] The San Francisco Chronicle called it a "stellar debut".^[22] It was shortlisted for the 2018 World Gourmand award for Best Indian Cookbook.^[10][23]

Khan was approached by Brian McGinn, producer of Chef's Table, to be the first British chef profiled on the series.^[11][7] Filming started in London and India in July 2018 with Zia Mandviwalla directing.^[11][14] Khan recalls Mandviwalla, who was born in Mumbai and lives in New Zealand, "did not ask me pointless questions about my husband and marriage, I did not need to explain what my mother meant to me, she got it."^[11] The programme first aired in February 2019. Khan is the first British chef to be featured.^{[3][4][11][13]} The series' sixth season's theme is "the journey home".^[13] The season, which included Khan's episode, was nominated for an Emmy in the outstanding documentary section.^[24] According to Bloomberg it became difficult to get a reservation at Darjeeling Express after the series aired.^[25]

Business Insider named her number 1 on their 2019 list of "100 Coolest People in Food and Drink".^[1] Danny DeVito offered to invest in an expansion.^[7]

Khan closed the restaurant The Darjeeling Express in March 2020 during the coronavirus pandemic.^[26] By the end of the year she had reopened in Covent Garden in a 120-seat space serving tasting menus.^[7][27] Khan has also been outspoken about the importance of her presence and the restaurant's in the cultural and social landscape in Europe.^[28][29]

On Sundays, when Darjeeling Express is closed, Khan offers free use of the premises to women who are aspiring chefs and restaurateurs who would like to host supper clubs.^[11][30] When Khan gave up the Soho space, she arranged with her landlord to allow Imad Alarnab, a Syrian refugee who had been running a pop-up restaurant, to use the space for the remainder of the lease.^[7]

Khan's restaurants support a non-profit, Second Daughters Fund, which encourages families in India to celebrate the births of second daughters by sending packages of sweet treats to be shared with neighbors.^[4][14]

In July 2019, to mark her 50th birthday, Khan traveled to Northern Iraq to open an all-women cafe for survivors of ISIS at the Essyan refugee camp.^[1][7][31]

In 2022, she was appointed the Chef Advocate for the United Kingdom (a WFP Goodwill Ambassador) by the UN World Food Programme.^[32]

Khan is married to Mushtaq, an academic.^[3] According to Khan, he is not a fan of her food, preferring simple dishes and finding hers too complex.^[3] The couple have two sons.^[13][3][11]

In May 2022 she was made an Honorary Fellow of the Queen's College, Oxford.^[33][34]