is it possible to reduce $HALT_{TM}$ to $E_{TM}$? I am wondering, if it is even possible: is it possible to reduce $HALT_{\text{TM}}$ to $E_{\text{TM}}$?

$HALT_{\text{TM}}=\{\langle M,w\rangle\mid M\text{ is a }TM\text{ and }M\text{ halts on input }w\}$

$E_{\text{TM}} =\{\langle M\rangle\mid M\text{ is a TM and } L\left(M\right)=\emptyset \}$

I tried to look online, in summaries and handouts and even in several books (including Sipser's book), but couldn't find any indication on it.

is it possible? if so, how?
Nice question.
No, you cannot reduce $HALT_{\text{TM}}$ to $E_{\text{TM}}$, although you can reduce $HALT_{TM}$ to $\overline{E_{\text{TM}}}$. Here we are talking about Turing reduction.
$HALT_{\text{TM}}$ can be reduced to $\overline{E_{\text{TM}}}$
Let $\langle M,w\rangle$ be such that $M\text{ is a }TM\text{ and }M\text{ halts on input }w$.
Now let us construct a TM $M'$. On an arbitrary input $u$, $M'$ will check whether $u$ is the same as $w$.

if it is not, $M'$ will run forever.
if it is, $M'$ will simulate $M$ on input $w$. Then once $M'$ halts, $u$ is considered as accepted.

We can make the transformation from $M$ to $M$ be algorithmic, thanks to the existence of a universal TM machine.
It is routine to verify that $\langle M,w\rangle$ is in $HALT_{TM}$ if and only if $M'$ is in $\overline{E_{\text{TM}}}$.
$HALT_{\text{TM}}$ cannot be reduced to $E_{\text{TM}}$
For the sake of contradiction, suppose $HALT_{\text{TM}}$ is reduced to $E_{\text{TM}}$ by reduction $f$. Then $\overline{HALT_{\text{TM}}}$ is reduced to $\overline{E_{\text{TM}}}$ by reduction $f$ as well. Since $\overline{E_{\text{TM}}}$ is Turing-recognizable, so is $\overline{HALT_{\text{TM}}}$. Since $HALT_{\text{TM}}$ is Turing-recognizable, we have $HALT_{\text{TM}}$ is decidable, which is known to be false, however. This contradiction shows that $HALT_{\text{TM}}$ cannot be reduced to $E_{\text{TM}}$.