I am in NO WAY an expert on the nutrition subject but my gut tells me that a diet of just chicken and carrot is just not covering the nutritional pyramid for doggies. I think you did mention vitamins although I myself would prefer to have them get all their nutrients from their food.
But as I said before I am sure that others on the list know much more than I as far as what is acceptable and healthy. I myself generally feed my 2 a premium quality dry dog food for the most part and some times cook for them.
Hope I helped even a little.
Tina