J.,
It sounds like your little girl is really frustrated with something. Is it possible the child she is pushing has done something to cause your daughter to react? Maybe the other child is making faces, sounds, or words to your daughter that she doesn't like and this is her way of stopping that.
Your daughter may also be struggling with a lack of communication. I suggest checking out your local library and looking for baby sign language books. These are easy signs that you and she can do together to help ease her communication struggles.
Lastly, when you get home from work spend 15 or 20 minutes just playing with your daughter, and let her lead the play. I don't know if you do this already, but the two of you have been separated for a very long day and she needs to know that she is the first thing you think about when you both arrive home. Yes, she does need to learn that you have tasks to accomplish and after your 20 minutes of uninterrupted play set a timer and let her know you need to work for 5 minutes and when the timer goes off you will play or sit with her for another 5 minutes.
While cooking dinner putting her in her highchair is a fabulous idea and a great way for her to feel included in the task of cooking.
Good luck.
-C..