Difference when using backticks in double quotes and single quotes in Bash

Why to the following two executions differ in output? (I need double quotes for variables in my eventual command)

$ sudo su -c "echo `cat /root/root_file`"
cat: /root/root_file: Permission denied
$ sudo su -c 'echo `cat /root/root_file`'
Yay, highly classified content!

Bash performs a series of expansions before the command is executed (sudo in this case). In the first case, the command substitution is performed because it's in double quotes -- it executes as you, hence the error. In the second case, the single quotes prevent expansions, so the backticks are not substituted until root's shell is running.

See also https://www.gnu.org/software/bash/manual/bashref.html#Quoting

In the first example the backticks are evaluated by your shell (as you and not root). Try with

sudo su -c "echo `whoami`"

In the second one with the single quotes ' the whole string echo `cat /root/root_file` is passed to the root shell.