How to make double-struck capital letters italic (e.g. U+1D53C)?

You can do it with XeLaTeX or LuaLaTeX. It can also be done by faking slant also with pdflatex, but this would need a whole set of virtual fonts.

\documentclass{article}
\usepackage{unicode-math}

\setmathfontface{\slantedmath}{Latin Modern Math}[FakeSlant=0.25]

\begin{document}

$ \slantedmath{}$

\end{document}

In pdflatex, borrowing from Shear transform a "box"

\documentclass{article}
\usepackage{amssymb}
\newsavebox{\foobox}
\newcommand{\slantbox}[2][.3]
  {%
    \mbox
      {%
        \sbox{\foobox}{#2}%
        \hskip\wd\foobox
        \pdfsave
        \pdfsetmatrix{1 0 #1 1}%
        \llap{\usebox{\foobox}}%
        \pdfrestore
      }%
  }
\begin{document}
$\slantbox{$\mathbb{E}$} = mc^2$
\end{document}