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It's a slang for somebody fat. 子 does not carry a specific meaning it is more a character with grammatical function to nominative

the commenter's username (i'm guessing they mean 大胖子, feel free to google translate)

This is the correct.

I was first called this by a Chinese classmate from Beijing with a biting sense of humor, when I was at university in Tokyo.

We got on really well, to be clear. :)

Hanging out with him was actually how I got started with Mandarin, probably why I chose this username.

I remember when first learning Mandarin coming across a phrase '你发福了' which literally compliments someone on blessings (i.e. having become more wealthy) but idiomatically means you gained weight.

I really like this one. It's delightfully cheeky. :)

胖 (pàng) means fat, vs 盘 (pán), which means plate.

Quite alright! We have to make mistakes to learn!