Preheat oven to 375° F. Butter a large pie pan or medium baking dish and set aside.
In a saucepan, whisk the arrowroot powder with water. Gently heat until it begins to thicken. Add the milk and heat until thickened, whisking frequently. Stir in the salt and dash of paprika.
Sautée the onion, celery, and garlic in 2 T butter until tender. Add mushrooms and saute until most of the liquid has evaporated.
Stir the drained tuna, peas, sautéed vegetables, and pasta into the milk. Pour into the baking dish and top with the grated cheese.
Bake 25-30 minutes in the preheated oven, or until the cheese is melted and lightly browned on top.