Well, this is a somewhat hidden feature, but minitoc provides for the \adjustmtc command, that is meant for starred chapters etc., i.e. for the case that the mtc counter is off by one.
This command can be 'misused' to provide the minitoc for the following chapter.
The \adjustmtc[n] command is documented on page 34 of the current manual of minitoc. The optional argument holds the number of minitocs to be incremented (default n=1)
\documentclass{book}
\usepackage{minitoc}
\usepackage{blindtext}
\setcounter{minitocdepth}{3}
\setcounter{secnumdepth}{3}
\dominitoc
\begin{document}
\tableofcontents
\markboth{}{}
\clearpage
\adjustmtc
\minitoc
\chapter{First}
\section{Foo}
\blindtext[2]
\subsection{FooFoo}
\blindtext[5]
\subsubsection{FooFooFoo}
\blindtext[5]
\chapter{Foobar}
\end{document}
Please note: The \setcounter... statements are just for filling up the minitoc etc. with a amount of entries -- they are not important for the functionality.

Update (with some the \cleartoevenpage code from `jfbu's answer!)
This works now automatically, i.e. \chapter adds the minitoc on the even page before.
\documentclass{book}
\usepackage{xparse}
\usepackage{minitoc}
\setcounter{tocdepth}{5}
\usepackage{blindtext}
\setcounter{minitocdepth}{3}
\setcounter{secnumdepth}{3}
\makeatletter
\let\latex@chapter\chapter
\newcommand*\cleartoevenpage {%
\clearpage
\if@twoside \ifodd \c@page \hbox {}\newpage
\if@twocolumn \hbox {}\newpage \fi
\fi \fi
}
% Automatic addition of minitoc
\RenewDocumentCommand{\chapter}{som}{%
\def\@@chaptertitlefortoc{#3}%
\IfValueTF{#2}{%
\def\@@chaptertitlefortoc{#2}%
}%
\IfBooleanTF{#1}{%
\latex@chapter*{#3}
}{%
\cleartoevenpage
\markboth{}{}%
\ifnum\value{chapter} = 0
\adjustmtc
\fi
\minitoc
\latex@chapter[\@@chaptertitlefortoc]{#3}
}
}
\makeatother
\dominitoc
\begin{document}
\tableofcontents
\chapter{First}
\section{Foo}
\blindtext[2]
\subsection{FooFoo}
\blindtext[5]
\subsubsection{FooFooFoo}
\blindtext[5]
\chapter{Foobar}
\blindtext[5]
\section{Foo}
\blindtext[5]
\chapter{Foobar other}
\blindtext[5]
\section{Other Foo}
\blindtext[5]
\end{document}
chaptercounter,\minitoc, decrement thechaptercounter. But maybe there is a right way to do it? – Norman Ramsey Feb 06 '16 at 17:53memoirclass has anopenleftoption. maybe that could be useful. – barbara beeton Feb 06 '16 at 18:10