In East Chilly Field, the intended route across the ice is to bark at the penguin and follow it. It will choose a route of ice "blocks" that are guaranteed to be safe to step on. The penguin can choose different paths, and stepping off of this path means you break the ice and fall into the lake, starting over... except there's actually a little more to it than that.

The blocks that are safe to walk on are actually always the same, regardless of whether or not you bark at the penguin or what route it takes. Here is an image showing which blocks are safe and which are not:

https://drive.google.com/file/d/17xY3SC1qucf2cDt3WiMeiYEzSyBNExJi/view?usp=drivesdk

Blue squares mark a block that will always break. Every other block is always completely safe (and, of course, the penguin is marked too). It's possible to memorise this and save a little bit of time- you don't have to wait for the penguin, because you can just go. Alternatively, if you're not as confident, you can follow the penguin up until a point where you're certain you can finish the rest of the path on your own.

Also of note: on the other side of the ice, there's a penguin family right next to the block(s) you have to exit the lake from, which can help a little.

Hope this helps someone a little in the future ^^