You need to enable the stdout pipe, which otherwise defaults to $*OUT, by setting :out. So:
my $proc = run("uname", :out);
my $stdout = $proc.out;
say $stdout.slurp;
$stdout.close;
which can be shortened to:
my $proc = run("uname", :out);
say $proc.out.slurp(:close);
If you want to capture output on stderr separately than stdout you can do:
my $proc = run("uname", :out, :err);
say "[stdout] " ~ $proc.out.slurp(:close);
say "[stderr] " ~ $proc.err.slurp(:close);
or if you want to capture stdout and stderr to one pipe, then:
my $proc = run("uname", :merge);
say "[stdout and stderr] " ~ $proc.out.slurp(:close);
Finally, if you don't want to capture the output and don't want it output to the terminal:
my $proc = run("uname", :!out, :!err);
exit( $proc.exitcode );