Melt the butter in a large saucepan over medium heat. Add the onions and sauté, stirring frequently until the onions are soft and slightly browned, about 5 minutes.
Add the flour, salt, pepper, and parsley flakes. Whisk well, and cook, whisking nearly constantly, for about 2 minutes.
Slowly add the chicken broth, whisking as you pour it into the saucepan.
Increase the heat to medium-high heat and bring the mixture to a boil, stirring often. Let the mixture boil for about 2 minutes.
Reduce the heat to medium heat and add the leftover mashed potatoes. Break up the chunks of mashed potatoes using a potato masher or a fork.
Once the potatoes are combined, add the milk and stir well.
Once the milk is heated through and a soft simmer has begun, add in the cheddar cheese and stir until the cheese is completely melted.
Remove from the heat and serve with additional shredded cheddar cheese for topping.