If you follow my answers, I really love to answer with "yes, nature loves to do all sorts of awesome things," and then link to some strange creature that does things you'd never expect possible.
This is not one of those answers.
The temperatures you describe are simply not accessible in the way you seek. They can't be achieved as an organ.
The first issue is the vacuum. One challenge with things that are as cold as 1K is that a single collision from a gas particle can raise its temperature well beyond 1K. When we get atoms down to below 1K, we do so in a vacuum chamber. Now this is not a "suck on the soda straw" type of vacuum. Atmospheric pressure is 760 Torr, where Torr is a unit of pressure. A human sucking on a soda straw can get around 500Torr. That's nothing. Get down to 21 Torr, and water will boil at room temperature.
The kinds of vacuums they use to get below 1K start on the order of 0.000000001 Torr and go down from there. At that point, gases stop acting like gasses, and start acting like little billiard balls bouncing around. When you learn about turbomolecular pumps, they are described as being less like a fan that pumps air out of your chamber and more like a carefully crafted set of baseball bats designed to strike individual atoms and push them towards the high pressure side where you can suck them out.
All sorts of strange things happen at those pressures. Anyone who works with ultra high vacuums like that knows the aphorism: everything outgasses. A single fingerprint can prevent you from reaching your goal pressure for weeks, simply by the oils in the fingerprint turning to gas, forcing you to pump them out.
You run into all sorts of fun issues constructing these things. You can't make them out of normal steel. Hydrogen can actually pour through mild steel, ensuring you never pump down. Modern vacuum aparatus are made of stainless steel. Not just any stainless, but a very particular alloy called 304L, and reasonably thick as well. So your "organ" now has to contain a stainless steel chamber at least 0.1" thick (quarter inch is common), made to the high standards of modern steel. It also needs to be produced in a way that doesn't leave a bunch of cells inside the wall, like how skin is a bunch of dead cells. It then needs to be cleaned with a whole bunch of chemicals: acetone and isopropyl alcohol are common.
We usually then bake vacuum components. Yes, we want to get to low temperatures, but we have to hit high temperatures first to help bake out anything that will outgas later. We might target 120C, well above the boiling point of water. Not enjoyable for an organ!
Once we assemble it, we have to pump it down. For this we tend to start with Turbomolecular pumps that spin at 90,000rpm, and then work our way down to exotic things like ion pumps and cryo pumps. You tend to need to use several pumps in tandem as they each have their own strengths and weaknesses. Even for a small chamber, this will easily use up 500W or more. This tells us something about the size of the creature. Expending 300W continuously is more or less the limit of a human body, so this creature most certainly is bigger (or at least more energetic) than we are. This will call for expending at least 10,000 Calories every day, for many days in a row. If this energy was stored as fat, that's one heck of a weight loss plan, as you'll lose about 3 pounds of fat every day just trying to keep this thing going.
And we haven't even gotten it cool yet.
We tend to rely on gasses to cool things down. Liquid nitrogen is useful to get down to 78K. But getting beyond that requires other gasses. Typically Helium-3 is used when trying to get down to ultra low temperatures. Note I said Helium-3. There's two isotopes for Helium: H3 and H4. Helium 3 is better because it boils at 3.19K instead of 4.214K. So now we need to get some Helium. Helium is extremely rare because it's small enough to simply leave the atmosphere once it gets there. Most of our helium comes from Uranium decay, and the helium gets trapped in natural gas, and its mostly H4. Most of our H3 comes from bombarding Lithium-6 with neutrons in a particle accelerator, releasing H4 and tritium, then storing the tritium until it decays to produce H3.
So our creature needs a particle accelerator, as well as a stainless steel chamber and a turbo pump and an appetite of a Michael Phelps!
Even H3 only gets you down to 3.19K the rest is up to you. You could try to go into the vacuum of space and expose the chamber to the void, but even the background radiation of the universe is too hot, at 2.7K. So you need something more exotic. Atoms being driven towards 1K and beyond are typically suspended in carefully managed magnetic traps, and then cooled with lasers. Now cooling with lasers is one of the neater tricks science has figured out. You've dealt with doppler shift, right? The train sounds higher pitched as it comes towards you and then lower pitched as it moves away? Well they abuse that. They have these lasers tuned such that the laser usually does not excite any of the atoms. However, if the atoms move towards the laser, the doppler shift is enough to cause the laser to apply pressure to it, gently pushing it the other way.
So you need a stainless steel chamber, perfect cleanliness a high speed and exotic pumping system consuming 3 pounds of fat a day just to keep it spinning, controlled magnetic traps, and lasers to cool things. Might as well strap the lasers to a shark, just for good measure.
And for what? There's not much to be gained from 1K and below. Almost anything you do with it raises its temperature. Scientists who push below 1K aren't looking to do anything with it. They're looking to learn something about the world around us. You only need a handful of these exotic contraptions around the world. An organ would mean each individual has one of these things. Even the scientists admit there just isn't enough of a use for 1K to go anywhere.
So I want to say "anything's possible for nature." But when you look at that laundry list of things, doing it in an organ is just not a likely path. Instead you'd do it exactly the way we did it -- in non-organic chambers crafted intentionally and with great purpose. And we'd only need a few of them.