In a small sauce pan, combine the carrots, onion, and enough chicken broth to cover. Bring to a boil and cook until very tender. Drain.
Prepare the mashed potatoes. Place chopped, peeled potatoes and cover with cold water. Add salt, pepper, and about a tablespoon of butter. Bring to a boil and cook until very tender but not falling apart, about 20 minutes.
Combine the milk and half-and-half in a coffee mug and heat in the microwave for about 1 minute, until just warmed through.
When the potatoes are tender, drain but reserve a little of the cooking liquid. Add about 3 tablespoons of the cooking liquid back into the potatoes, and mash by hand at first. Once the biggest chunks are broken down, turn on the the electric mixer and add the milk/half-and-half mixture until the desired consistency is reached. Note: you will not use all of the milk mixture.
Add a tablespoon of butter and salt and pepper to taste to the potatoes. Mix well.
Mash the cooked carrots by hand until mostly smooth. Add the carrots into the potatoes and mix well, adding more of the milk liquid if needed. Taste to check for seasoning.