Place milk, pumpkin puree, cinnamon, nutmeg, and cloves in a medium saucepan and stir together. Simmer for 15 to 20 minutes.
Fill a large pot with water and bring to a boil. Once the water comes to a boil add a generous amount of salt (2½ – 3 tablespoons) and then the pasta.
Boil pasta for 10 to 12 minutes or until al dente, stirring occasionally. Drain pasta and set aside.
Melt butter over medium heat in a medium saucepan. Sprinkle flour over melted butter and whisk together. Allow mixture to cook for 2 to 3 minutes. Stir in thyme.
Whisk pumpkin-milk mixture into flour mixture and continue to whisk until fully incorporated. Allow mixture to simmer until thick enough to coat the back of a wooden spoon, 3 to 4 minutes.
Place shredded cheese into a mixing bowl and toss together with lemon juice.
Stir cheese into the béchamel sauce and stir until cheese just melts. Season with salt and pepper. Reduce heat to low and gently stir pasta into the sauce, ½ cup at a time until all of the pasta has been added and is fully coasted in sauce. Adjust seasonings and serve immediately.