Conn. Gen. Stat. § 4-2: When nominations are to be made.

Except as otherwise provided by law, all nominations to be made by the Governor for appointment to office by the General Assembly or either branch thereof, and all appointments to office to be made by the Governor which require the advice and consent of the General Assembly or either branch thereof, shall be made on or before May first of the year in which the General Assembly is in session, except that, in any case where a vacancy results from death occurring on or after said May first but prior to the adjournment of such session, the Governor may make a nomination or appointment to fill such vacancy as soon as possible thereafter and prior to such adjournment and if such vacancy is in the office of a judge, the nomination shall, notwithstanding the provisions of section 2-40, be referred to the committee on the judiciary immediately on receipt thereof from the Governor.