Queen Elizabeth Cake is a Canadian white cake, made rich by the generous addition of dates and nuts.
It’s frosted with a butter, sugar, coconut and cream frosting.
What’s unusual about the cake is that after being frosted, the cake goes back into the oven for a bit to lightly brown the frosting.
Some versions have you boil the topping first before putting it back into the oven but it’s not necessary. Some versions have you only boil the topping, and *not* put it back in the oven. But in all variations, the topping is cooked in some way.
It is not clear where the name for Queen Elizabeth Cake actually comes from.
Apparently the recipe might have been sold, for 15 cents a copy, as a fund-raiser during the Second World War, and as Queen Elizabeth, the Queen Mother, was very popular in Canada and rallied Canadians during the war, it may have been named in her honour.
The recipe was also sold as a fundraiser after the war, by Brownies  .
It definitely appeared in war-time cookbooks during the 1940s. It re-appeared in Canadian cookbooks in 1953, for the coronation of Queen Elizabeth II, and since then, due to its ease, has been a staple at country fairs since then. It is also known in the UK and in the States, though it’s not as ubiquitous there as it is in Canada. It appeared in the States by the early 1950s, being introduced by Canadian connections. 
It is very closely related to Lazy Daisy cake (which doesn’t have the dates) and Sticky Toffee pudding.
In the 1950s and 1960s, it was even used as a wedding cake , often tiered. 
Literature & Lore
“Today Cook’s Corner has thrown caution and calories to the winds. We have three great desserts for you to try. Our guest cook is Mrs. W.B. (Jessie) Hutchinson of 820 Park Ave. The Hutchinsons like to go to Canada to fish and often stay at a lodge where the food is one of the main attractions. During their last visit, the chef served a cake that merited a recipe request. The recipe turned out to be a Queen Elizabeth Cake, supposedly the favorite dessert of England’s current monarch… but you don’t have to be royal to enjoy it!” — Heard[sp?], Peg. Cook’s Corner Column. Tyrone, Pennsylvania, USA: Tyrone Daily Herald. 7 July 1982. Page 3.
“…Jaymie began her Queen Elizabeth Cake, turning the sticky dates into a newer glass bowl, boiling the kettle and pouring one cup over the dates and baking soda, which fizzed up…. I think you do this to soften the dates, so they blend well with the moist cake batter, Jaymie said….. [Jaymie rose] to take the cake out of the oven. She had already boiled the odd “icing” — it was made of brown sugar, coconut, butter and one other ingredient she had hoped to guess at; she hoped “top milk” meant cream — and poured it over the cake. It pooled, so she got a nutpick out of the drawer and poked holes in the top, letting the brown sugar mixture ooze into the cake. She then stuck the pan back in the oven, watching it carefully so she could tell when the coconut had browned slightly.” — Hamilton, Victoria. A Deadly Grind. New York: Berkley Publishing Group. 2012. pp 74 – 76.
 “A successful ice cream tea was held recently by the 2nd Lethbridge Brownie Pack in the Susie Bawden auditorium. Brownies served at tables attractively centred with miniature toadstools surrounded by Brownies. Receiving at the door were Brown Owl Mrs. T. M. Allen and Tawny Owl Mrs. R. B. McKenzie. Plants grown by individual Brownies were sold, as well as recipes for the Queen Elizabeth cake. An added attraction was a large pantry table. Conveners were Mrs. Doug Card and Mrs. R. C. Kimber.” — In and Out of Town Column. Lethbridge, Alberta, Canada: The Lethbridge Herald. 1 June 1953. Page 14.
 “I got this recipe at my shower 45 years ago,” writes Bella Nicholson of Morden, adding the recipe was sold as a money making or charitable purpose [Ed: sic] for 15 cents.” — Simon, Illana. Recipe Swap column: Queen Elizabeth cake clearly a royal treat. Winnipeg, Manitoba, Canada: Winnipeg Free Press. 11 October 1995. Page 26.
 “The recipe for Queen Elizabeth cake, submitted by Mrs. Elsie M. Wells, Lebanon Route 20, seems to be different. It was given to her by her great aunt, Mrs. Margaret McDonough, who is a native of Lebanon County but has lived in Nova Scotia, Canada, for the past 18 years.” — My Favorite Recipe column. Lebanon, Pennsylvania: Lebanon Daily News. 18 February 1954. Page 36.
 “The bride’s table was laid with white lace over blue centered by a Queen Elizabeth cake flanked by tall blue tapers in crystal holders…” — Miss Pegues Becomes Bride Of Clyde Winston Walters. Galveston, Texas: Galveston Daily News. 30 December 1959. Page 5.
 “The bride’s table was laid with lace cloth over pink and centered with a satin bow and lilies of the valley. It held a three tiered Queen Elizabeth cake topped with a miniature bride and groom.” — “Mr., Mrs. Sharp Return From New Orleans Honeymoon.” Baytown, Texas: The Baytown Sun. 2 December 1966. Page 9.