How can I use newcommand/DeclareMathOperator to this?

You can parametrize the variable part.

\documentclass{article}
\usepackage{amsmath,bm}

\newcommand{\bmu}{\bm{\mu}}
\newcommand{\wwmti}[1]{%
  \widehat{\widehat{#1(\Theta_i)}}%
}

\begin{document}

$\wwmti{\mu}+\wwmti{\mu_1}+\wwmti{\bmu}$

\end{document}

If you have access to the mtpro2 package -- or, at least, its "lite" subset, it'll give you some really wide single-hat and double-hat "accents".

\documentclass{article}
\usepackage{amsmath,bm}
\usepackage{newtxtext}    % Times Roman clone text font
\usepackage[lite]{mtpro2} $ Times Roman clone math font
\newcommand{\dwidehat}[1]{\widehat{\widehat{#1}}}

\begin{document}
$\dwidehat{\bm{\mu}(\Theta_i)}$, 
$\dwidehat{\mu_1(\Theta_i)}$, 
$\dwidehat{\mu_2(\Theta_i)}$
\end{document}