Freeze ice to far below 0 Celsius?

Most of the cooling from ice comes from the melting anyway. That's why ice makers, which don't freeze as cold as freezers, are still useful.

It takes 334J/g to turn ice at 0C into water at 0C, but only 2.03J/g to warm ice by 1C. So to halve the amount of ice you use to get the same cooling you'd need to freeze it to around -150C. If you're going to do that you may as well use the liquid nitrogen directly, because that's the most likely way to get ice at -150.

If you use dry ice or liquid nitrogen in the kitchen, you need to look into the precautions (and regulations if it's commercial) that you need to take.

You should be able to cool the sauce by transferring it to a bowl sitting in an ice+water bath (this cools faster than sitting it in a bowl of ice because of contact area). If that's not enough, here are some ideas that can be used as well

This is done with wine and cocktails - it might work with your sauce:

• make a batch of the sauce (or a base without cream/egg etc.)
• freeze it
• use the frozen sauce (base) to cool the final sauce. It may dilute ingredients you couldn't freeze but this will be much less noticeable than diluting with water.

Sitting a bowl inside another bowl, with the gap full of ice+water, then frozen, should get you a very effective cooling device for moderate quantities. It should scale to a litre or so per vessel if there's plenty of room for ice in between. The inner container should have decent thermal conductivity and/or thin walls - a stainless serving dish would be ideal.

Another approach is to use an ice/salt bath for freezing point depression. This can be used to make ice cream so gets really quite cold. You could precool a bowl in that and then put your sauce in it.

Don't freeze ice then. There's a couple of solutions where something with a good amount of heat capasity is popped into a freezer and used to cool things - the generic names for these seem to be drink chillers. They are made of materials like metal or granite which you throw into a fridge. It'll cool, and when added to a drink will absorb heat from it, and cool down your drink without added ice.

The way to store a small quantity of dry ice for a week without spending a lot of money on a freezer that will actually keep it below sublimation temperature, or having a cylinder of the pressurized liquid and a "dry ice maker attachment" for the cylinder to make some when you need some, is to buy a larger quantity of dry ice and store it in a well-insulated container. A Dewar flask would be the best option; a large amount of styrofoam is less ideal, but can be good enough. Nested foam coolers (ie, a foam cooler with dry ice inside another foam cooler) can hold it for a while, but details will depend on variable factors. You do need to consider the ventilation of the area where it is stored, as it is possible to fill an area with the heavier-than-air gas and manage to kill yourself, or others, or pets.

As for the practicality of using it in cooking, marginal at best (and dry ice, in particular, may cause taste effects from dissolved CO2 in the cooled product (ie, sparkling .vs. still water.)

Your martini, for instance, could be much more easily super-cooled by storing the gin or vodka in the freezer - I suspect the vermouth might freeze (not a martini fan, but most wines will freeze in a common residential freezer, while most distilled spirits won't.) Perhaps you could make vermouth cubes.

Chris H's answer has good details regarding latent heat and use of ice/salt mixtures (I upvoted)

In commercial kitchen environments large quantities of liquids that require rapid chilling may be stirred with a "cooling paddle" - the "secret" to which is generally a water-based (or simply water) filling that is pre-frozen to make ice, and a large surface area to rapidly melt the ice, pulling the latent heat of melting from the hot liquid, without mixing in water.

Pouring it into a hotel pan in a shallow layer, and setting the hotel pan in another hotel pan full of ice is another effective approach.