Remove the skins and heads from the shrimp and set aside. Devein the shrimp by cutting into the outside of the shrimp and removing the usually black string-like 'guts'. Chop each shrimp into roughly 3 - 4 pieces.
To make the stock, first if you like, cook the shrimp heads and shells in a little butter (around ½ tablespoon/7g) for a minute or two. in a medium pot over a medium heat. You can skip this if you are short of time, but it helps get a bit of extra flavor from the shells. Add the water to the pot then cover and bring to a boil. Reduce the heat slightly and let the mixture simmer for around 30-35 minutes, preparing the rest as it simmers.
Meanwhile, peel and finely dice the onion and carrot.
Warm 1 tablespoon of the butter in another medium pot over a medium-high heat until it bubbles. Add the shrimp pieces and cook for a minute or two until cooked through. Remove with a slotted spoon and set aside.
Add the remaining butter, let it melt then add the onion and carrot. Cook until both soften and the onion is translucent, around 10-12 minutes, but without letting them burn.
Add the tomato paste and stir through. Let it cook a minute then add the brandy. If you like, you can warm it ahead of time and set it alight as you add to quickly burn off the alcohol and give a bit of extra flavor. If you are unsure, you can just cook it a minute or two longer to cook off the alcohol - it should become almost syrupy.
Next add the white wine and let it reduce, stirring regularly, by at least half.
By this point, the stock should be about ready. Strain the shrimp stock with a fine strainer and add it to the onion-carrot mixture. Add the rice, thyme and season with salt and pepper. Cover and simmer for around 10 minutes until the rice is cooked through.
Reserve 4 small stacks of shrimp (a little less than half) and add the remainder back into the soup.
Blend the soup well and add the cream (see notes). Then, pass through a fine strainer (or cheesecloth-lined wider strainer). Press the solids well so that you get a good amount of them coming through as they help thicken the soup.
If needed, return the mixture to the heat to warm through. Take care not to let it boil.
Place a small stack of the cooked shrimp in the middle of each bowl (use smaller, flatter bowls if possible). Pour the soup around the shrimp so some still show through then serve.