You don't "handle" other people, you handle your own thoughts. I know how easy it is to be upset by other people's judgments; I do it myself. But I've also been learning that the best I can do is the best I can do, and the decisions I make are the right ones if I made them thoughtfully, with the best intentions toward all affected parties.
That doesn't mean other people must agree with me or approve of my decisions. But it's not their business. And the more I learn to keep my judgments out of other people's choices (and it is a learning process that I undertake with real attention), the more immune I gradually become to their opinions of my choices.
You know, usually when somebody's judgments, or even an innocent statement or facial expression, hit us wrong is when we are not completely sure we did the right thing. So their comments tug on our "Did I really? Was I wrong?" doubts. They might be just below our mental notice, but our feelings are more in touch with our sense of doubt, and so we feel annoyed or agitated and assume it was what the other person said or did.
It isn't. It's usually what we think ourselves, but have buried. It's usually a judgment we still hold against ourselves, in spite of our rationalizations. It's usually the doubt we feel that we haven't quite put to rest.
So those upsets actually provide good information and self-knowledge, and you might feel much more peaceful toward this other woman if you recognize what the information is offering you. It may mean, for example, that you still have some processing/grieving to do over the loss of your dog. It may be rooted in an earlier life memory or regret. But it really is more about you than about your co-worker.
I've done a great deal of work on figuring these things out – I use a process called The Work that has been tremendously de-stressing, freeing me from my thoughts about other people's judgments. And it can actually be fun sometimes. Here's a link if you are curious. You can watch videos of other people doing The Work, and download free resources to try it yourself: http://thework.com/thework.php