What is a cold email? It's an Initial message sent to a potential supervisor.
The subject of the e-mail: Prospective PhD / Postdoc Fellow 2023
Here is a much better approach in the sample inquiry below:
______________________________
Dear Professor X (name the professor)
I am completing my undergraduate studies and have read your work on (X subject). It has inspired my interest in a research career in your field, particularly in the (X specialty area—named specifically). I was especially inspired by your article(s) (books) (NAME THEM SPECIFICALLY). I plan to apply for the Ph.D. program at your university, and am writing to ask about the possibility of working with you if I am admitted.
(Then list your research experience, your credentials, etc. and attach a curriculum vitae)
________________________________
Do not write things like: “I am considering applying to your university” or “Your university is one of the places I am thinking of applying.” This is arrogant. You will be very lucky to get a position at any university, since students are extremely expensive and time-consuming to train. In your statement of purpose when applying, you must definitely state exactly why you want to study at that particular university. Do not make it seem like you are the one making the decision (as true as this might be). Every letter, every inquiry must be personalized and specific to each institution where you are making inquiries.
In the end DO YOUR HOMEWORK. A lazy approach to graduate education is never going to be successful. Any institution that accepts you for the Ph.D. will be paying you tuition plus a living stipend in the U.S.A. This can have a total value of $60,000 per year or even more. Take this seriously. Do make inquiries of professors with whom you hope to work, but don’t think that if you just send out a lot of boiler-plate letters that you will get any kind of response.
Summary
Your Cold email should be short and have a maximum of 3 paragraphs:
1. Introduction
2. Research Interest and Professor’s article or book(s) you’ve read
3. Express your desire to be supervised by such a renowned Professor
Finally, attach your CV, Research Proposal & wait.