How to use corn starch instead of flour to thicken soups.
I disagree that cornstarch wont remain thickened when cool. I use it to thicken soups/sauces all the time and have no problem with the soup not being thick enough upon reheating, although depending on what kind of soup you're making, it might have to be whisked well to bring it back together when reheating. I use 2:1 cornstarch to liquid to make a slurry.
What kind of soup are you talking about? You thicken soup all different ways, depending. Usually you don't use either flour OR cornstarch. If you cook beans in vegetable soup, they thicken it. Same goes for rice or pasta (and GF pasta does exist). For a milk-based seafood chowder, I thicken with a little instant mashed potato. A cream soup might be thickened with flour, so buy some GF flour. I make a Polish soup that is chicken broth thickened with nothing but mashed potato and seasoned with nothing but salt and fresh dill. And other posters have described thickening with a puree of vegetables.
If you want glutton free, just be a bit more discerning on who to invite.
I don't know much about gluten free.
When thickening soups with cornstarch (Chinese egg drop or hot and sour come to mind), I use about a third of a coffee mug full of cornstarch. I add liquid until its about 1/2 to 2/3 full.
Liquid should be cool to warm, not hot. You can use water or broth (turkey or chicken as in your recipe would be good).
Bring soup to boiling (this is key). While boiling, slowly stream cornstarch mixture into pot, stirring well.
Stop adding the cornstarch mixture when you get desired thickness.
If still too thin, repeat until you get desired thickness.
Cornstarch has its strongest thickening power only when the liquid is boiling.
As magiesmom mentions, the thickness will break when cooled.
To use cornstarch as a thickener, you mix it 1:1 with water until you have a slurry. Add this to your liquid at the end of cooking until you achieve the thickness you want.
That said, cornstarch is not an ideal thickener for a soup like this. A pure starch tends to gel in liquid giving a thick consistency. What you want in the mulligatawny is a creamy consistency. If you want to avoid using a roux, xanthan gum is a thickener that gives low-fat liquids a full-fat mouthfeel. You can also simply introduce fat in the form of heavy cream or coconut milk and reduce the soup until it is thickened and creamy.
in that recipe they are using the flour in the second stage almost like a roux rather than as and added thickener - I think whisking in cornstarch at the end to thicken will not have the same richness of the flower that is browned with the spices and veggies and fats. Google turns up some results for "gluten free roux" I have never tried so I cannot recommend one but I would look to replicate that part of the process as much as possible - or just sub a gluten free flour there.
I don't like cornstarch to thicken soup because it will not maintain thickness when it reheats.
You can instead thicken the soup by taking out a little before you add the meat but have added everything else and blending it with a traditional or immersion blender. IMO this is a far preferable way to thicken soup anyway.