Yes you can carve on ice. I do it every day of my life when I am training. It's easy. If I were you I would read the Carving on Ice: Equipment or Technique thread that ended last year about this time, and read the recent Carving: Perception vs Reality thread that just recently died down. Both are excellent sources of not only how to carve, but what a "carved" turn actually is.
Now, I have never skied Stowe, so forgive me for my ignorance, but I always thought that a good majority of the front 4 were completely ungroomed (never seen a groomer), bump/crud runs full of natural terrain features... I also just gave their trail map a once over, and it turns out that only one of them is even partially groomed. So now I pose the question to you: How do you carve in bumps? I can scarve bumps, and I can GS carve over top of bumps... but actually laying out respectable carves in bumps is stupid - unless you don't like the fillings in your teeth.
I think by carve you actually mean turn (god knows what kind - but a turn none-the-less). So now we have narrowed this down to you can turn on the front 4 (assuming you mean all 5 of them), but when you hit real ice elsewhere you can't set an edge into it to TURN on it. At this point I am doubtful that you are carving, so lets stick with that terminology for now.
I suggest you read those two threads. I will post the links again in this thread:Who is Carving: Perception vs. RealityCarving On Ice: Technique or Gear or What?
In these threads there were some very useful posts by myself and others. You will find a lot of information that you can easily apply to your skiing and carving on ice/groomers. Be mindful of what you are reading as well. Not all advice given is "good" advice if you're looking to improve upon what you are already doing or may not be doing.