In a medium saucepan, melt Butter (3 tablespoon) and stir in All-Purpose Flour (2 tablespoon), Water (2 cup), Salt (1 tablespoon), Ground Black Pepper (1/8 teaspoon) and Tarragon (2 tablespoon).
Add the Milk (1 1/2 cup) and blend well. Bring to boil over medium heat, stirring until thickened and smooth.
Stir in the Cheddar Cheese (1 1/2 cup), cook over low heat, stirring constantly until cheese melts.
In a lightly greased 9×13 inch baking dish, layer half the Potato (6 cup), Onion (2) and Carrot (4) mixture. Top with half of the cheese sauce.
Repeat. May sprinkle the top with a bit of Fresh Parsley (to taste) (or add some extra cheese on top).
Bake, covered for 30 min or until they are tender. Remove cover the last 10 minutes of baking to become a golden brown.