In other words, even if you do everything humanly-possible to avoid probate, which is to say you use a revocable living trust (and actually fund it, by the way!), you have updated beneficiaries on your life insurance, retirement accounts, annuities, etc., you use joint ownership effectively, and you use TOD/POD features if available, someone will still need to file documents with the probate court upon your passing.
The most important document to file, regardless of how many assets (if any) have to go through probate, is the estate tax return. The return needs to list everything in your name when you died, including probate and non-probate assets. The court will use this return to determine if there is any estate tax liability to attend to.
Now, having said that, please keep in mind that there is an enormous difference between filing a few documents with the court and going through a full-blown probate process (maybe 10-12 months of administration, even if everything goes smoothly). So it is still worthwhile to consider some probate-avoidance maneuvers you can make before you pass.
So, to sum up, you can only minimize Connecticut probate upon death. You cannot avoid it entirely.