Cook the pasta: Bring a large pot of salted water to a boil. Cook pasta until just shy of al dente. Reserve 1/2 cup of the pasta water, then drain.
Blend the sauce base: In a blender, combine cottage cheese, milk, Dijon, garlic powder, onion powder, smoked paprika (if using), salt, and pepper.
Blend until completely smooth and glossy. This should take 45–60 seconds.
Warm the sauce gently: Pour the blended mixture into a large skillet or pot over low heat. Stir continuously until warm and slightly thickened, 2–3 minutes.
Do not let it boil.
Add the cheeses: Stir in cheddar and Parmesan a handful at a time, letting each addition melt before adding more. Keep the heat low to prevent graininess.
Adjust consistency: If the sauce feels too thick, splash in warm milk or reserved pasta water, 1 tablespoon at a time, until it coats a spoon.
Stir in yogurt (optional): Off the heat, whisk in Greek yogurt for extra tang and protein. Taste and adjust salt and pepper.
Combine with pasta: Add the drained pasta to the sauce and toss until every nook is coated.
If needed, loosen with another tablespoon of pasta water.
Finish and serve: Top with chives, cracked pepper, a dusting of Parmesan, or toasted breadcrumbs for crunch. Serve hot.