Do you get any nutritional value from cooking with veggies but not actually eating them?

Answer

If you cooked broccoli (or vegetables) in a soup, but didn’t eat the vegetable and yet ate the soup, you would get some nutrition because some vitamins from the broccoli would now be in the soup. If there was rice in the soup, the rice might even absorb a small amount of those vitamins as it absorbs the water from the soup. But in your example, there would be very, very little – or no — nutrition benefit (from the broccoli) if you eat only the chicken.