Eftir að hafa — after having done

After having done

You already have two pieces: finite eftir að clauses (eftir að ég borðaði) and the perfect infinitive hafa + supine from the modal lessons (hann gæti hafa farið). Put together they give the compact written pattern:

eftir að hafa + supine — after having done

The subject is not repeated — it is understood from the main clause:

Both are correct; the compact one is what newspapers and formal prose prefer. When the eftir-phrase opens the sentence, the main clause inverts (V2): Eftir að hafa borðað *fór hann að sofa.*