PowerShell has hidden re-quoting logic when arguments are passed to external utilities.
What you're seeing is bug in Windows PowerShell (that has since been fixed in PowerShell Core):
In short, "cn=\`"Jim Carter\`"" is passed to an external executable as cn=\"Jim Beam\", which unexpectedly turns into two arguments - note how the argument is not double-quoted as a whole, even though it needs to be.
Workarounds:
As part of a distinguished name in Active Directory, component values (such as Jim Beam for field cn) usually do not need double-quoting, even if they contain spaces. Therefore, perhaps the following will work (depending on the behavior of the executable you're invoking).
someExternalTool.exe "cn=Jim Beam"
If you do need the executable to ultimately see string literal cn="Jim Beam", you must use --%, the stop-parsing symbol:
someExternalTool.exe --% "cn=\"Jim Beam\""
If you want the executable to see cn=\"Jim Beam\", i.e., with the " chars. \-escaped, use someExternalTool.exe --% "cn=\\\"Jim Beam\\\""
Note that use of --% then precludes the use of PowerShell variables as part of the same command.
In PowerShell Core, your original argument, "cn=\`"Jim Carter\`"" is passed as
"cn=\"Jim Carter\"" - i.e. with the necessary double-quoting as a whole - so the problem with the unexpected argument splitting wouldn't arise, and your target program would see cn="Jim Beam" after performing its own argument parsing.
If you need the target program to see cn=\"Jim Beam\", you'd have to use
"cn=\\\`"Jim Beam\\\`"" - or, using single-quoting,
'cn=\\\"Jim Beam\\\"'
As an aside: The need to \-escape " chars. to pass to external programs - in addition to PowerShell's own escaping - should itself be considered a bug, but it's a longstanding one that probably won't be fixed so as to maintain backward compatibility: see this GitHub issue.
'cn="Jim Carter"'? By using single-quote characters around the whole string, the inner double quotes should be preserverd. Otherwise, try and figure out what the external tool uses as escape characters as @AdminOfThings commented.tool.exe --% cn="Jim Carter". I can't be of much help without knowledge of the tool itself I think.