Can American Math. Monthly be used to publish hard research?
Of course read the description on the AMM web page about what sort of thing they publish. https://www.maa.org/press/periodicals/american-mathematical-monthly
The Monthly's readers expect a high standard of exposition; they look for articles that inform, stimulate, challenge, enlighten, and even entertain. Monthly articles are meant to be read, enjoyed, and discussed, rather than just archived. Articles may be expositions of old or new results, historical or biographical essays, speculations or definitive treatments, broad developments, or explorations of a single application. Novelty and generality are far less important than clarity of exposition and broad appeal. Appropriate figures, diagrams, and photographs are encouraged.
Your Ph.D. research should be published in a research journal. If some part of it is of general interest (as above) it could then be written up for publication in the Monthly. Of course that version of the result will contain much more background and motivation than the research paper.
You did not explain why you want to publish there since there are so many good research journals. I can hardly see your motivation.
AMM can publish hard results, but only if they are of substantial general interest and only if the proofs are short and not too difficult to follow. I have seen new proofs of classical and rather difficult theorems. One of my favorite is:
A. Dold, Simple Proof of the Jordan-Alexander Complement Theorem The American Mathematical Monthly 100 (1993), 856-857.
Unless as a part of your research you proved something very elegant and of general interest, something that can be explained in a detailed manner in a few pages, your result will not qualify for AMM. What you think about your result does not really matter. It has to be appreciated by others.
Publications in AMM will rarely be regarded as original. Most of the papers give another look at the material that is known already.
Definitely they publish serious research. For example, this paper
MR0379852 L. Zalcman, A heuristic principle in complex function theory. Amer. Math. Monthly 82 (1975), no. 8, 813–817
has 175 citations as recorded on Mathscinet. (And 351 on Google Scholar). And many other such examples can be given. As I understand their criterion is that the paper is a) of sufficient interest to the broad audience, and b) does not require much background. There are plenty of serious research papers satisfying both criteria.