I have been curious about the Kozhukkattai. It appears to be in the genre of 'momo' a Nepali/Tibetan dumpling. The concept of dumpling exist in several civilizations. But I just can't imagine in South India, where idli and dosai were the norm, somebody would decide to make a dumpling.
My uninformed theory is Kozhukkattai came to the tamil region from elsewhere.
I am seriously searching. But spl kozhukkattai like " Mukkuruni" in Madurai ( but the moorti was found only during Tirumalai Nayak period), Pillayarpatti did exist but from when is the question.
Praying to lord Ganesa to throw some light on it. May be Saraswathi Mahal library can throw some light. Not in the list given by Avvayar or Arunagirinathar.
Pillayar answered the questions. Somebody Played ganesa pancharatnam of Adisankara ( Adisankara from Kerala , then Chera Country, Tamil speaking,. Sankara is variously dated but even if you take the 9th C,) then Kozhukkattai aka Modakam was in 9th C in Tamil country.
Ganesa Pancharatnamstarts with - Mudaa karaatta modakam - One who is having a modak in hand.
Pillayarpatti Pllayar is also holding something ,isterious - variously stated as Sivalingam, fruit and Modak.