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 the Russian empire in the early 19th 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, usually about 6 months. Traveling the "other way" (not through the Empire but through America and the Atlantic Ocean) was in fact sometimes faster, because ships travel faster than sledges.

The 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. The speed on land widely varied: it depended
on the infrastructure and on the ability to change horses and to buy supplies.
It also depended on season: in summer you have many rivers to cross (including very large ones), while in winter they can be crossed on sledges on ice.

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).

EDIT. On October 7 1852 a Russian diplomatic mission started from StPetersburg to Japan (by sea). Japan has a common boundary with the Russian far East possessions. They arrived to Japan in the beginning of August 1853.
So it took 10 months by sea. The mission returned by land. They started on August 15 1854 in the Russian far East, and arrived to StPetersburg on
25 of Februaty 1855. 6 months by land. The travel is described by the secretary of the mission, the famous Russian writer Goncharov in his book
<a href="https://en.wikipedia.org/wiki/Frigate_%22Pallada%22">Frigate "Pallada".</a> 

In 1904/5, during the Russian Japanese war, the Russians sent a fleet (consisting of steam ships) from the Baltic to the far East to fight the Japanese. The fleet departed on  15 October 1904 and arrived to Tsushima straight in Japan on 28 May 1905. At that time a railroad crossing the empire from West to East already existed. So we see that steamships in great hurry made this is 7 months while
a sailing ship (perhaps not in so great hurry) made it 50 years earlier in 10 months.