Preheat oven to 375° F.
Bring a large pot of water to a boil. Add the egg noodles and cook according to package directions, being careful to not overcook them. Drain and rinse in cold water.
Meanwhile, melt the 3 tablespoons of butter in a large skillet over medium heat. Once melted, add the onion and celery and cook, stirring occasionally, until softened, 4-5 minutes.
Add the flour to the skillet and stir for 30 seconds to cook off any raw flour taste. Then stir in the chicken broth and milk, followed by the salt, pepper and dried thyme. Cook, stirring often, for 2-3 minutes, until the sauce thickens.
Remove the pan from the heat. Stir in the frozen peas and carrots, tuna, shredded cheddar cheese and cooked noodles. Transfer the mixture to a 9x13-inch baking dish.
To make the topping, in a medium bowl, stir together the panko, grated Parmesan and 1 tablespoon melted butter. Sprinkle evenly over the casserole.
Bake, uncovered, for 15-20 minutes, until casserole is hot throughout and topping is golden brown. Serve, garnished with chopped fresh parsley if desired.