Chapter 146, sec. 9-222 of the Connecticut General Statute (see here) reads,

When a vacancy occurs in the office of first selectman or in the office of selectman it shall be filled within thirty days after the day of its occurrence by the remaining members of the board of selectmen. Said remaining members may appoint one of themselves to fill a vacancy in the office of first selectman, if they so desire, and shall then fill the ensuing vacancy in the office of selectman as herein provided. If such a vacancy in the office of first selectman or of selectman is not so filled within thirty days after the day of its occurrence, the town clerk shall, within ten days thereafter, notify the elective town officers enrolled in the same political party as the first selectman or selectman, as the case may be, who vacated the office, or all elective town officers, if such first selectman or selectman who vacated the office was not enrolled with a political party, and it shall be filled by such elective town officers within sixty days after its occurrence. Any person so appointed shall serve for the portion of the term remaining unexpired or until a special election called as hereinafter provided upon petition of a number of electors of such town equal to five per cent of the names on the last-completed registry list thereof, but not fewer than fifty such electors. Such petition shall be filed no later than fifteen days after the appointment by the remaining selectmen or such elective town officers, as the case may be. Such a special election shall forthwith be called by the town clerk upon the filing of such a petition with him and shall be held in accordance with the provisions of sections 9-164, 9-450 and 9-459. The term “town officers”, as used in this section, shall not include state representatives or town officers who serve on town boards whose members are not all elected at one town election for the same term.

1. Selectman resigns.
2. Remaining members of the Board of Selectmen have 30 days (until January 3, based on the date of Liz’s resignation) to fill the vacancy with anyone they choose. The board currently has two Democrats and two Republicans, so minority representation rules don’t come into play. The board could agree on a Republican, a Democrat, or someone unaffiliated with either party.
3. If the BOS can’t agree on a replacement, the Town Clerk, starting January 4, has ten days (until January 13) to notify the elected Republican town officers of their right to appoint Liz’s replacement.
4. The Republican town officers then have 20 days (until February 3) to appoint someone (20 days because the language says the appointment must be made within 60 days of the resignation).
5. As soon as Liz’s replacement is appointed, voters have 15 days to submit a petition requesting a special election. Five percent of the electorate (approximately 170 voters) are needed to call a special election.