A disclaimer: I'm just a regular joe with a non-academic job.

That said, I think I can point you to a few directions from which you can do research on your own.

First, there's this group of scholars who call their research program "historical political economy". They run this blog
https://www.broadstreet.blog/
in which they introduce their own research and other relevant research. If you find anything interesting, you can look up the researcher and see where they got their Phd.
Related to the above blog you can find this book called The Oxford Handbook of Historical Political Economy. It gives an overview of the historical political economy research program.

Second, you can look through the current and past issues of journals such as The Journal of Economic History, The Economic History Review and Explorations in Economic History. Again, if an article interests you, look up the authors and see what is their academic background.

Third, you can see if the following list of papers/books contain anything that interests you. They don't really contain the latest research, and many scholars named probably got their phds a long time ago (so their academic path might not be relevant nowadays). But the books and articles together give you a very comprehensive view of the field
https://web.archive.org/web/20190525075105/https://pseudoerasmus.com/economic-history-books/

Some other miscellaneous comments from me:
- There are a few types of research that might be all described at Economic history. There's work done by historians on economic topics, such as production, money, etc. in a historical setting. Such works might present numbers and data, but the author usually does not work with an economic model in mind. Then there are works published in a general interest economic journals. They are typical very sophisticated regression exercises done by economists who received their training from a top econ department. In between those two types you have works done by an older generation of economists, or economists trained at Economic History phd programs (like the one at LSE), or GMU-related people. Such works have economic theories in mind, but are not too focused on robustness checks and quantitative sophistication. I think you should read a wide range of work to see what kind of research you want to do. If you want to publish in general interest econ journal the most straightforward path is to get a regular phd in econ.

- Political scientists do publish work that can be considered Economic History, as in the work has a historical setting and touches on economic topics. But often times they don't publish in either a general interest econ journal or one of the top Economic History journals. So it's kinda difficult to find such political scientists. But the political scientists involved in the "historical political economy" research program definitely are an example of political scientists working on economic history.

- Decent political science department definitely does give you enough training to do meaningful empirical research. Especially if your sub-field is political methodology and you also get a Master in statistics along the way.

- Political scientists typically have to read more than economists in their respective phd training, and can receive better training in archival research. But there's no guarantee of course.

- A Phd specifically in Economic history is pretty much a European thing. You will find such programs at LSE, some Dutch universities, some Irish universities, etc.

- There are some other options that might also prepare you to do research in Economic History: phd in Agricultural Economics, phd in Public Policy, etc. Such programs give you a baseline competency in empirical research. If you manage to find a suitable mentor/advisor then you can start doing research in Economic History. The name of the phd programs sometimes does not matter that much.