In a 4-quart saucepan, melt the butter and saute the onion and carrots until both are slightly tender, about 5 minutes. Whisk in the flour and cook for 1 minute. Add the potatoes, milk, and dissolved bouillon cubes. Cook over medium heat for 20 minutes, until the potatoes are very soft and some of them have begun to become very mushy. Add the half-and-half, salt, and pepper. Remove from heat and set aside.