I stumbled on this recipe in The New England Economical Housekeeper ©1845.
A very good pie may be made of carrots in the same way that you make pumpkin pies.
Yes, that's all they had for the recipe. So, I decided to hunt down a few more.
Carrot Pie. from The American Housewife and Kitchen Directory ©1869
Scrape the skin off from the carrots, boil them soft, and strain them through a sieve. To a pint of the strained pulp put three pints of milk, six beaten eggs, two table-spoonsfu of melted butter, the juice of half a lemon, and the grated rind of a whole one. Sweeten it to your taste, and bake it in deep pie plates without an upper crust.
The New England Cook Book's recipe is similar but slightly different. Original publication 1836
Scrape three good sized carrots, boil them till very tender. Then rub them through a sieve, and mix them with a quart of milk, four beaten eggs, a piece of butter of the size of half an egg, a table spoonful of lemon juice, and the grated peel of half of a one. Sweeten it to your taste. Bake it in deep pie plates with an under crust and rim.
I could find recipes from other sources but they were all similar to the ones above.