It is a strange way, of course to measure the SIZE of an empire in terms of the TIME of travel from end to end; at different times different means of transportation were available, and it also depended on the season.

But if one accepts this strange criterion, perhaps Russian empire in the early
19s century is hard to beat: it stretched from Alaska and California to Poland. The only ways to travel from the far east to the capital
was either around the whole Asia, Africa and Europe by sea, or by sledges across Siberia. The land travel from the Far East to St.Petersburg took several months in the best case. Traveling the "other way" (not through the Empire but through America and Atlantic ocean) was in fact sometimes faster, because ships travel faster than sledges.

Mongolian empire at its largest extent was almost the same size, excluding Alaska and California. It stretched from Korea to Poland, but they probably had better postal service at that time.

When Batu and Subutai conquered Poland, the Great Khan Ögedei died in Karakorum.
(This was the capital of the empire, about 2/3 of the  way from Poland to Korea). He died in December 1241. Batu and Subutai left Poland in the "late spring" 1242.
I suppose they left immediately after receiving the news, and that the news was passed in the fastest possible way).