Pre-heat the broiler. Place a large pot of salted water over high heat to boil.
Place a large skillet over medium heat with 2 tablespoons butter. Add the onion, celery and carrots; season with salt and pepper and cook until tender but not caramelized, 5-6 minutes. Add the potatoes, thyme, juice of the clams and 1/2 cup of water to the skillet and bring the liquids up to a bubble. Cover the pot, reduce the heat to medium and simmer until the potatoes are tender, 8-10 minutes.
While the veggies are simmering, drop the pasta into the boiling water and cook to al dente, according to the package directions. Just before the pasta is ready, reserve a large mug of the starchy cooking water. Drain the pasta and return it to the pan it was cooked in.
While the veggies and pasta are cooking, place a medium size saucepot over medium heat with the remaining 2 tablespoons butter. Sprinkle the flour over the melted butter and cook for 1 minute. Whisk the milk and 1 cup of starchy cooking water into the roux and bring up to a bubble. Add the clams to the sauce; season with salt and pepper and simmer until thickened, about 1 minute. Reserve warm.
When everything is ready, toss the cooked veggies and prepared sauce with the cooked pasta and transfer to a large casserole dish.
Sprinkle the breadcrumbs and grated cheese over the surface of the casserole and brown under the broiler.