Much of European ancestry today is from the Anatolian Neolithic Farmers who originated from Anatolia, which is now Turkey, which is technically in the Middle East and starting spreading into Europe around 7,000 BC, mostly replacing and intermixing with the Western Hunter Gatherers who were in Europe before them.
They form the bulk of the ancestry of Southern Europeans such as Greek, Italian, Spanish, etc. Way even before that, the first humans to ever cross into Europe most likely did it from the Middle East. If you are talking about the first empire from the Middle East to expand into Europe.