In the end, neither side could claim the win, though it was Everton and their supporters left celebrating an unexpected point and Tuchel left searching for answers to a growing list of issues.
A trip to Wolves on Sunday afternoon is next in the diary before a Carabao Cup quarter-final at Brentford next Wednesday.
Chelsea (3-4-3) Mendy; Azpilicueta (Chalobah 79), Thiago Silva, Rudiger; James, Loftus-Cheek (Barkley 65), Jorginho (c), Alonso (Saul 65); Ziyech, Pulisic, Mount
Unused subs Arrizabalaga, Bettinelli, Christensen, Sarr, Kante
Scorer Mount 70
Booked Loftus-Cheek 45; Jorginho 59
Everton (4-4-2) Pickford; Kenny, Keane (c), Holgate, Branthwaite, Godfrey; Doucoure, Gomes (Gbamin 70); Iwobi, Simms (Dobbin 62), Gordon (Onyango 90)
Unused subs Begovic, Lonergan, Coleman, Allan, Gbamin
Scorer Branthwaite 74
Booked Holgate 21; Gomes 64; Dobbin 77
Referee Michael Oliver