Persian empires (plural) might be a better way to look at it. There were multiple dynasties across centuries, and while there were consistencies, there were big differences too. If you want to read about ancient Persia (beginning 500 BCE) through to the era of the Byzantine Empire, I strongly recommend A History of Zoroastrianism, by Mary Boyce. It is a 3 volume work and is absolutely definitive. I know of no finer nor comprehensive investigation of Zoroastrianism, and it is a heavily historical work. Reading that will give you about 1,000 years of Persian history, and I'm sure her bibliography for the work will have some great leads for further reading.

As u/Write_What_I_Like said, "the Persian empire" is a bit of a misnomer. There was usually something that could be described as a Persian empire from c.550 BCE - 1979 CE. Of course, a lot of that stretches the definition of either "Persian" or "Empire," and most people are thinking of something ancient when they say Persian Empire, usually the Achaemenid dynasty that invaded Greece in the early 5th Century and was eventually conquered by Alexander the Great.

Mary Boyce's A History of Zoroastrianism is a (unfinished series) great series, though only volume 2 really deals with Persian rule. However, it's also very specialized and niche, specifically focusing on religion. Boyce's work is also a bit outdated in the eyes of more recent scholars, so I always recommend pairing it with something more recent like Prods Oktor Skjærvø's Introduction to Zoroastrianism (which has the nice benefit of also being free from Harvard).

For an introduction to that first Persian Empire, I usually recommend Ancient Persia: A Concise History of the Achaemenid Empire, 550-330 BCE by Matt Waters.

From Cyrus to Alexander: A History of the Persian Empire by Pierre Briant is THE book on Achaemenid Persia and has been for the last 20 years (the price tag matches the reputation, I'm afraid).

The Persian Empire: A Corpus of Sources from the Achaemenid Period by Amelie Kuhrt is an excellent resource which compiles primary sources and commentary.

Persians: the Age of the Great Kings by Lloyd Llewellyn-Jones is one to look out for. It isn't published yet, but is slated for early next year.

The World of Achaemenid Persia: The Diversity of Ancient Iran edited by John Curtis and St. John Simpson is a good starting point for getting into some specific aspects of Achaemenid history.

Encyclopaedia Iranica is a free, online, academic encyclopedia that's absolutely invaluable for any part of Iranian history

Ancient Persia by Josef Wiesehöfer covers the whole span of Persian history from the rise of the Achaemenids, through the Hellenistic and Parthian Periods, and on to the Sassanid dynasty and the Arab-Islamic conquest.

The Persians by Maria Brosius also covers the Achaemenid, Parthian, and Sassanid Empires but focuses more on cultural, rather than political, history.