It all depends on your point of view. If you consider the Khazarian Khanate an Empire (which can be consider an equivalent) and you have a broad definition of Jewish people, then that's your answer.

The Kazharian Khanate was a steppe state that occupied the region from Crimea (West) to the Aral Sea (East) and the Caucasus (South) and existed from 7th to 10th centuries CE. As the the title Khan suggests, we're talking about a Turkic nation, but its population was diverse, including Finno-Ugrian, Slavic and Jewish communities.

The practice of several religions was legal on the realm, so Christianism, Islamism, Judaism and Paganism were common. The original religion practiced by the head of state and the higher social strata would probably be Tengriism, but somewhere around the 8th century, they converted to Judaism. They weren't ethnically Semitic, but they practiced Judaism.

I'm sorry I can't give you more detailed information on the subject. In the meantime, I think I will read this and maybe I can add something more.

There were the original kingdoms of Israel and Judea, of course—the last of which was conquered by the Neo-Babylonians in the 6th century BCE. In the 2nd century BCE the Maccabees led a Jewish revolt against the Seleucids, which created the independent Hasmonean kingdom that was eventually absorbed into the Roman Empire.

Over the following centuries the Jews—like the Armenians, Georgians, and Arabs—were in the middle of the battleground between Rome and Persia, and were alternately courted and oppressed by both sides. When the Persians conquered Syria in the cataclysmic Persian-Byzantine war of the 7th century CE, they set up a short-lived autonomous Jewish state; but this was reconquered by the Byzantines soon after.

Aside from the “Jewish people”, Judaism as a religion was adopted by several non-Jewish states attempting to assert their independence from neighboring Christian, Zoroastrian, and Muslim powers. These included Helena of Adiabene (in modern Kurdistan) in 30 CE; the Himyarite king Yusef Dhu Nuwas in pre-Islamic Yemen; and most famously, the Turkish Khazars in 8th-century southern Russia.

So, I think by "empire" you mean "political entity", since the definition of empire is a bit complicated, but is more narrow than what it seems you're interested in from your text.

There've been a few Jewish countries in the past. Already mentioned is the Khazar Khangate, which was probably Jewish to some extent, but how much so is not entirely clear. Another one already mentioned is the rather obvious kingdoms of antiquity, the Judean and Israelian kingdoms (though describing these as "Jewish" is a bit iffy), and the Hasmonean Dynasty. Even more obvious is the modern state of Israel.

But a more interesting one not discussed in depth yet is the Himyarite Kingdom of the Arabian penninsula. Himyar was a kingdom in the region for several centuries in pre-Islamic Arabia. In the 5th century, the ruling family converted to Judaism, and restricted Christian practice in their lands. However, they'd been fighting a turf war with the Byzantine Empire for a while, and when they went on the offensive, attacking Christian areas, the Byzantines got a second wind and with the assistance of Christians in Ethiopia, destroyed the Himyar kingdom, and ended the brief Jewish political rule in Arabia.