I stumbled upon this post while trying to do something similar;
I ended up implementing the following new commands, which, though naive, worked smoothly enough for my document:
\documentclass{amsart}
\newcommand{\ellipse}[1]
{
\begin{center} \tikz\node[draw, fill = blue!20!white, ellipse]{#1}; \end{center}
}
\newcommand{\hide}[1]
{
\texorpdfstring{\protect\phantom{\hspace*{5em}#1}}{#1}
}
\newcommand{\csec}[1]
{
\section{\hide{#1}}
\vspace*{-5ex}\ellipse{#1}
}
\usepackage{amsfonts, amsmath, amssymb, amsthm}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{shapes, snakes}
\usepackage{hyperref}
\begin{document}
\section*{Note}
To make a circled, unnumbered section, you could define a new command \verb`csecun` just like \verb`csec` but using the \verb`section*` command.
\csec{First Section}
\csec{Second Section}
\end{document}