Most recipes for carrot soup I encounter require a final stage of blending/whizzing to get the creamy texture. Eg:
Can the blender/whizzer be avoided? How must the recipes be modified in that case?
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One way to modify the recipes (but you won't get the smooth texture of the original recipes) is to cut the onion and carrot a lot smaller than the original recipe ask for. You want to almost dice it. When you are frying the onion and carrot in the pan, do it for like 1-2 min rather than the whole 5 minutes. And finally you will need to cook it in the stock for much longer than the recipe asks. You want to bring the stock to a boil then let it simmer covered to allow the carrot to become really soft. Then use something with a big flat surface(a large wooden spoon or something similar) to help you mash the rest against the side of the stock pot. The results will come out a lot lumpier than the orginal recipes but some texture and "unblended" bits of carrot is actually pretty good in carrot soup.
In case you find yourself without a blender, but DO find yourself with a pressure cooker, follow the recipe sauteeing the ingredients, then just add maybe 1/2 cup of water or stock, and pressure cook for 5 mins under full pressure (if you have the timing kind) or 1-2 whistles (if you have the whistling kind). Once this cools and you can open it, you will see that your carrots are VERY soft and can basically be mashed up into puree. You add the stock and whatever else of the ingredients at this point and cook, but it won't need as much cooking as it says in the recipe.
The onion will not mash as much (though still pretty well), so that I would reccommend chopping a bit smaller to help get a smoother end result.
You can also do this with cauliflower for a nice cauliflower soup, or even potatoes (though I would double the pressure cooking time for potato). Should work with other things as well.
I don't think you need to blend anything. If you like good chunks, skip the blending step.
If you don't like big chunks, cut the carrots and other veggies in smaller pieces. I have no experience with carrot soup, but if possible, try to mash it with a potato masher, before you add the liquid. If this isn't possible (because the veggies are still too hard), put the liquid in, let it simmer as instructed and instead of putting it in a blender, use the potato masher then. You will have some smaller chunks left, but nothing large, normally.
You could buy carrot puree sold in a can. Most of your recipes called for a weight so you could use that accordingly from the can's size (also in lbs. for that specific can) or water down to your desired thickness.
If you are just against buying (or using) electric machines, maybe you can find a food mill (wikipedia link) which used to be used for baby food and to make smooth soups and even potato mash before the machines became popular.
For a first try, if you do not have such a machine, you can use a sieve or colander over a pan or bowl and push the food through with a spoon or a something like that. This is much harder work but how hard depends on how well done your veggies are.
Both these methods require your carrots (and other veggies) to be well cooked, so that they fall apart when pushed, more so with the sieve method.
Creamy soups were invented before machines, so old methods were available before newer ones pushed them out of use.