Bring a large pot of water (5 or 6 quarts) to a boil with 2 teaspoons salt.
Drop spinach into the water and cook 1 minute; remove with a slotted spoon to a colander and drain. Put the spinach in a clean kitchen towel (careful, it's hot), roll it up and squeeze over the sink to remove as much water as you can. Chop spinach.
Parboil the pasta until halfway cooked, 7 - 10 minutes depending on the size and thickness of the pasta. Drain.
Stir together cream, garlic, salt, pepper, chicken, spinach and the cheeses in a large bowl; add drained pasta and mix together.
Smear the butter over a 1-quart gratin or baking dish. Transfer pasta mixture to the gratin. Bake 12 minutes or so, until the the cream is bubbling and edges of the pasta begin to brown. Cool 10 minutes before serving.