Does anyone get the joke here? Is it about Teto's ability to eat without getting any taiyaki on her face?

For one thing she seems to be noisier. And she's breaking bits off and eating them instead of biting into it. *shrug*

I think it's her peculiar way of breaking off the Taiyaki pieces... She eats the fins first (the ones without the filling in most cases), and then the head.

In a sense, she's eating the edges first.

if you cooking by self. You will get this joke.

Teto eat like prepare a fisk to cook.

First, remove any fish's fins. And then remove fish's head.

Now, your fish already to cook!!

Dammit, I was so close!

The sound of her breaking/snapping off the pieces (ブチッ) is getting progressively louder though.

Gott said:
[...]
Teto eat like prepare a fisk to cook.
[...]

It's the Swedish Chef!

I think it's crying in the final panel.