The first cake was baked with a single egg, before more were gradually added to the recipe over a two-month period.
Now Clara, six, does not have an egg allergy. And thanks to the daily consumption of the few types of nuts that did not put her at risk of anaphylactic shock, she is no longer allergic to nuts either.
Her younger brothers Grady, four, and Arthur, two, also showed signs of having the allergies and were treated the same way, under guidance of Dr Lack. They, too, are now allergy-free.
The new approach is based on a number of studies that saw development of a potentially fatal peanut allergy fall by more than 80 per cent in children under five by slowly introducing tiny portions of the food to them. Experts found that allergies can be overcome in as little as six months.
Food allergies are on the rise globally. A 2015 study by the European Academy of Allergy and Clinical Immunology found that the prevalence of food allergies on the Continent had doubled in ten years, while the number of people requiring hospital treatment for severe reactions increased sevenfold compared with the previous decade.
An estimated one in ten pre-school children in developed countries has a clinically diagnosed food allergy.
Ms Nixon Pompa's book, Allergy-Free Kids, is based on the studies carried out by Dr Lack, and disputes NHS advice that allergens should not be introduced to children aged under six months.
Current health guidelines state that when introducing solids to children, foods that may trigger an allergic reaction, including milk, eggs, wheat, nuts, seeds, fish and shellfish, should not be given before the child is six months old.
However, growing evidence suggests that high-risk children - those with severe eczema, egg allergy, or both - should start being fed these foods at four to six months. Other children should be introduced to them at three months.
The Government's Scientific Advisory Committee on Nutrition (SCAN), which advises Public Health England and the NHS, is currently reviewing its advice.
One of the trials influencing the change is Dr Lack's Learning Early About Peanut allergy (LEAP) study, published in 2015.
Children who were fed peanuts three or more times a week were 81 per cent less likely to develop a peanut allergy than those who avoided the food.
Dr Lack's subsequent study, Enquiring About Tolerance (EAT), published in March last year, found infants who had been given cow's milk, peanuts, sesame seeds, fish, wheat and eggs from three months old were 67 per cent less likely to have any food allergy.
Researchers found that the more eggs and peanut butter the child had eaten, the less likelihood there was of them developing an allergy when they became toddlers.
There was not a single case of anaphylactic shock, a severe reaction that can cause fatal swelling of the airways and suffocation.
However, it is important to consult a doctor before introducing an allergen.
Dr Lack warns: "Success is dependent on starting early, and not all food allergies can be improved. In some cases a child might have been desensitised. However, if the child stops eating the food, the majority become allergic again so it's not a cure."
Allergy-Free Kids: The Science-Based Approach To Preventing Food Allergies, by Robin Nixon Pompa, is published by William Morrow on May 18, priced £16.99.
