Fill a large pot with about 4 quarts of water. Bring to a boil then add a generous amount of salt (about 1-2 tablespoons).
Add cauliflower and carrots and boil until cooked through and easily pierced with a knife, about 7-9 minutes. Use a fine mesh sieve to transfer the veggies to a high-speed blender. To the blender add ¾ cup of milk and salt and blend until totally smooth.
Add the pasta to the boiling water and cook according to the package instructions. Drain and set aside.
Return the pot to the stove over low heat. Add the butter, remaining ¾ cup milk, mustard, and spices and stir until butter is melted and mixture is combined. Stir in the veggie puree and yogurt until combined and heated through, about 1 minute.
Turn off the heat and start sprinkling in the cheese in increments while stirring constantly until it has melted into the sauce. Add the pasta back to the pot and stir to coat.
Serve right away topped with lots of black pepper, if desired.