Can I substitute Carrot Juice for Tomato Soup, in 'Tomato Soup cake'? Do I need to adjust the recipe in other ways?
Grab some vegetable juice (V8 or something like it) and bring it to a gentle simmer. Put a teaspoon of corn starch in 3/4 tablespoon of water, stir the mixture rapidly for a bit, then stir that rapidly into the vegetable juice. Taste for salt. Bring it briefly to a rolling boil and then take it off the heat. It should coat the back of a spoon.
Keep stirring, occasionally as it cools, and you'll have something you can use. You can't directly substitute carrot juice for condensed soup, and simply thickening carrot juice in a similar manner as what I described above is going to give the cake strange sweet and wet spots after it bakes. I share Sharlyn's concern about the cake rising properly and evenly.
You could try a puree of carrot, thicken it a bit as described above, and go with that, but the minimal viscosity you need is probably that of vegetable juice after being thickened.