Rinse the fish fillet and remove all bones. Cut into small pieces.
Bring water to a boil in a medium saucepan. Add the potatoes, carrots, and broccoli, then cook for about 10 minutes. Add the fish and cook another 10–15 minutes or until the fish flakes easily. When cooked, drain and return the fish and vegetables to the saucepan.
In a separate small saucepan, melt the butter. Stir in the flour. Once mixed, add in the milk and stir constantly until a thin sauce is formed. Mix this sauce in with the fish and vegetables, stirring to combine.
Divide the fish and vegetables into a 6-cup muffin tin. Bake at 350°F for 30 minutes. Let cool before serving, and fork-mash if desired.