It is likely that the only way you can file a lien (before you have a judgment) is with an attachment, a prejudgment remedy that will require a court hearing and several weeks to get set up. It is also likely cost prohibitive as the company will neeed a lawyer admitted in Connecticut to handle the matter.
the limit for small claims court is $5000 and the comapny is allowed to represent itself. Tht will likely take several months before you could get a judgment. Once you had a judgment, you could file a judgment lien. Good luck
It is probably too late to get a statutory lien for the work performed on the property. Next time take action and see a lawyer sooner. Find out the small claims limit in your jurisdcition and consider that option. You should consult with an attorney in your area specializing in this type of matter; it is important to disclose all the facts and to explore your options. Be prepared to pay a consultation fee and know what it will be in advance.