If your distribution has the pgrep command installed (e.g. from the procps package), then you can use the command to find process IDs by pattern. For example:
# Find the sleep commands belonging to the current user.
pgrep -u $LOGNAME sleep
This command will find all matches belonging to the current user, whether the process was started in the current terminal or another, so it will work for your PuTTY use case.
You can test this out on a single TTY or PTY quite simply, too. Consider:
$ sleep 30 & pgrep -u $LOGNAME sleep
In this example, sleep is put into the background and Bash reports the Job ID and PID of the backgrounded job. Then pgrep finds the same PID in the process list, and sends the PID to standard output. Obviously, the numbers should match.
Note that if the current user has more than one sleep command running, then pgrep will correctly report multiple PIDs. For example, the following returns PIDs for background jobs
$ sleep 30 & sleep 30 & pgrep -u $LOGNAME sleep