From what I read from your code, your approach is the following:

read through the file "$file"
when a starting session is found
    open the file "$file" again, as file2
    read through file2 until the end of the session is found
[download]

I would approach the problem differently. I would try to read through the file only once, remembering the start and ending time for each session:

my %session_start;
my %session_end;
while( <FILE> ) {
    if( /$pattern/ ) {
        # Session starts here
        my $started_session= $1; # actually, you do the counting of pa
+rentheses
        my $session_start_time= $2; # actually, you do the counting of
+ parentheses
        if( ! $session_start{ $started_session }) {
            # That session really is new
            $session_start{ $started_session }= $session_start_time;
        } else {
            # That session should be new but we already know it
            warn "Line $.: Session '$started_session' already started 
+at $session_start{ $started_session }, but also (re)starts at $sessio
+n_start_time. See [$_]";
        };
    } elsif( /(\d+-\d+-\d+ \d+:\d+:\d+).*session:(\S{32})/)
        my $session_end= $1; # actually, you do the counting of parent
+heses
        my $ended_session= $2; # actually, you do the counting of pare
+ntheses
        # Session was seen here, or might end here
        if( ! $session_start{ $ended_session }) {
            # That session is unknown?!really is new
            warn "Line $.: Session '$ended_session' never started, but
+ ends at $session_end. See [$_]";
        } else {
            # That session was seen here
            $session_end{ $ended_session }= $session_end;
        };
    }
};
[download]

Once you have collected all the start and end times for the sessions (and refined your regular expressions so they only match the start/end lines), you just need to output the start/end hashes:

for my $session (sort keys %session_start) {
    print "Session: $session - $session_start{ $session } - $session_e
+nd{ $session }\n";
};
[download]

Sanity checking that every session that was started also ends, and that every session that ended also started is left to the user.