Melt butter in a large skillet over medium heat; add ham and brown sugar. Cook and stir ham until browned, about 5 minutes. Remove ham with a slotted spoon to a plate, reserving drippings in the skillet.
Stir cream of mushroom soup, evaporated milk, water, onion, salt, and pepper into the reserved drippings in the skillet; add potato slices and carrot pieces. Place a cover on the skillet, reduce heat to low, and cook mixture at a simmer until potatoes and carrot are tender, about 35 minutes. Return ham to skillet and continue cooking until ham is hot, about 10 minutes.