[elcaro]

Just for fun, I tried it in Raku.

  say qx< cal -hy >                                 # Command output as string
    .lines.skip.join("\n").split(/\n\n+/)           # Skip year & split groups
    .map({ |([Z] .lines.join.comb(22).batch(3)) })  # Single column structure
    .map({ |(.[2] ~ .[0].trim, |.[3 .. *]) })       # Month name on first row
    .grep(*.trim.chars).join("\n")                  # Join non-blank lines
    .subst(/\s+ ^^ \s\s+ 1»/, '  1', :g)            # Collapse weeks together

I was never going to beat APL... Though I imagine some clever Rakoons could probably find a few places where this can be golfed down.

[elcaro]

The first operation where I split to lines, then join, then split again... was annoying me. So now I've got this

  say qx< cal -hy >.lines.skip.batch(9)
    .map({ |([Z] .join.comb(22).batch(3)) })
    .map({ |(.[2] ~ .[0].trim, |.[3 .. *]) })
    .grep(*.trim.chars).join("\n")
    .subst(/\s+ ^^ \s\s+ 1»/, '  1', :g)