Preheat oven to 350 degrees F. Grease a 9x13 inch casserole dish.
Cook pasta in salted boiling water according to directions on box until just al dente. Reserve 1/4 cup starch pasta water before draining (just scoop out with a mug). Drain and set aside.
Meanwhile, brown the sausage over medium-high heat, breaking up into small pieces as it cooks. Remove from skillet.
Reduce heat to medium. Melt the butter in the same skillet and then whisk in the flour until it starts to bubble. Add the garlic and cook until fragrant. Whisk in the chicken broth a little at a time, followed by the pumpkin puree, milk, tomato sauce, spices, and thyme. Bring to a simmer.
Stir in the the spinach, mixing and letting wilt after each handful and then add the beans.
Mix the pasta, sauce, sausage, and pasta water together.
Spoon half the mixture into the pan, cover with 1 cup cheese, add remaining pasta mix and then remaining cheese. Sprinkle with fresh thyme sprigs.
Cover with foil and bake for 20 minutes. Uncover, turn oven to broil and cook for 3-5 minutes or until golden.