This is a very difficult question to answer, and indeed it would be impossible to name one person who is the very "best" critic or analyst of John Donne's verse. Donne has been the subject of extensive critical discussion for hundreds of years, especially during the nineteenth and twentieth centuries. Many would argue that the modern American poet and critic T. S. Eliot had a decisive influence in stimulating modern interest in Donne, but few would agree that Eliot alone is the "best" critic of Donne.
Fortunately, two extremely valuable resources now exist that make it possible to attain a very broad view of what hundreds of the best critics have had to say about Donne's poetry. One of these resources consists of the detailed annotated bibliographies of Donne criticism prepared by John R. Roberts. These bibliographies offer brief summaries of thousands of articles and books written about Donne. Thanks to the detailed indexes of Roberts' volumes, it is possible to trace very precisely what many different critics have had to say about practically every poem written by Donne. The following links will provide more information about Roberts' bibliographies:
Even more detailed than Roberts' volumes are the multiple volumes of the "Variorum" edition of Donne's poetry. The general editor of this edition, which is not yet complete, is Gary Stringer. This edition of Donne's verse provides extremely detailed line-by-line, even word-by-word summaries of what thousands of critics have had to say about Donne's poetry. The Donne Variorum edition is one of the great monuments of modern scholarship. For more information about this edition, please visit this link: