So the PLACE approach might be to say, to grumpy DD, something like "you seem to be cross, I wonder if you're a bit tired or (whatever you think might underlie it)" and carry on getting breakfast ready/ whatever. Later, find a moment to ask her what she thinks would be helpful to help her keep her temper.
My DD will throw herself around on the sofa, wailing (I'm sorry for the neighbours) when she gets up if she's still tired (and she keeps herself awake at night, but that's another story...). I tend to say that I think she's tired and doesn't feel good at the moment, then chat away about what I'm making her for her packed lunch. Not rewarding the behaviour but not punishing it either.
Generally in any situation you have to choose between building a relationship or controlling behaviour. The more you can build the relationship the better, because that's the basis of long term improved behaviour - but that means as often as is 'safe' ignoring behaviour you don't want.
Do you think the time outs are working? Or do you do them because you feel you ought to do something? Have you tried ignoring any outbursts, and noticing and praising the mornings when she is polite?
What is it triggering in you and your beliefs/expectations of how she should behave? I'm not saying that those expectations are wrong or unreasonable, but it is good to be intentional about expectations rather than just going along with those set by our parents. Is she able to control her feelings and deliberately choosing to be rude and upset you? Or is something she genuinely struggles with at the moment but will probably be able to control her feelings better in a year's time?
The most useful thing I've done is to let go of believing that I can control/ be responsible for DD's feelings. I can influence them to a certain extent, and I will stop her damaging herself, others or belongings (not that I've had to do that for a long time). But the best thing I can do for her is stay calm and not get dragged into an emotional reaction. It is more effective to notice and praise and encourage what I do want than try to stop what I don't want.
Don't sweat the small stuff. If she is generally kind and well mannered and wants to be 'good', then as she gets older she'll get more able to control the morning grumps. And you're practicing therapeutic parenting for the day when you'll REALLY
need it.
And yes, we all get angry and frustrated at times... it is also therapeutic to let your child know that you're human and have feelings. If I shout I make sure I apologise afterwards - I look on it as 'role modelling making a mistake and apologising for it'
which is a vital life skill 