"With whom am I talking?"
Translation:Je parle avec qui ?
When 1st group (-er ending) verbs are used in the interrogative form, the final -e of the 1st person takes an accent --> je parle -> parlé-je https://leconjugueur.lefigaro.fr/conjugaison/verbe/parler_question.html But this use is indeed very formal and has become extremely rare.
Many things on your sentence...
"Est ce qui" doesn’t exist. The only possible phrase is "est-ce que" + subject + verb.
Thinking about it, except with devoir and pouvoir (dois je, puis je), you very rarely use the inversion with "je"
Your sentence lacks the conjunction "with"
If you really wanted to use the "est ce" construction, you could say "avec qui est ce que je parle?". It is a bit heavy, but correct.