Chickens can peck a mouse or frog into pieces they can eat. Those beaks are dangerous weapons. Cooked carrots are not a challenge at all.
Mine like practically any vegetable cooked although they’ll eat a lot raw veggies too. I regularly toss them carrots peelings. That’s where you use a vegetable peeler to peel carrots and get long thin “shavings”. They like them that way. Carrots help darken the egg yolks.
I grow carrots and many other vegetables. Occasionally I’ll toss whole raw carrots I to them, the ones that are too twisted to be really useable. They’ll normally peck them apart and eat them, though they don’t go crazy over them. As I said, those beaks are dangerous weapons.
Chickens are not consistent. I’ve tossed them things that they are supposed to love, like cabbage leaves, and they sometimes ignore them. Eventually they usually learn to eat them but it is not always love at first sight.