A319 Captain reported high EGT after aircraft was deiced and elected to shut the engine down and return to the gate for maintenance action.

Narrative

FO's (First Officer) leg; but on the ground taxiing; so I as Captain was PF (Pilot Flying). Pushed back from the gate; de-iced; anti-iced; started engines; and began taxiing. During taxi; we got an ECAM message of high EGT on engine 1. Stopped the aircraft; checked the EGT; high 600s and climbing. Shut down the engine and returned to the gate. Informed the passengers and Flight Attendants. Called Dispatch and Maintenance Control. Maintenance checked their data and said the EGT was over limit for 15 seconds; and peaked at 788 Degrees C. My initial concern was that the de-icing crew had not completely cleared the engine inlet. Now my concern is that maybe we didn't have the engine anti-ice on. It seems unlikely that we wouldn't have turned it on. We talked about the icing conditions; the Anti-Ice ON; the de-icing; anti-icing; taxiway conditions; etc. But now I can't say with 100% certainty that we did.Taxiway conditions were okay; but not great. Snow over some ice. When we got the chime; my first action was to stop the aircraft and set the parking brake; verify the parking brake; and no aircraft movement. Only then did I look at the ECAM to see what the issue was. I'm sure this prolonged the time over the EGT limit. The other thing that initially confused us; and led to a couple of extra seconds over the limit; was the message itself. >ENG 1 EGT high. >IF unsuccessful: >Engine Master switch off. Our question: 'If WHAT is unsuccessful?' Must be to reduce thrust; but it's already at idle. Let's shut it down.