Boil the pasta: Cook pasta in well-salted water until just al dente.
Before draining, reserve 1 cup of starchy pasta water. Drain the pasta.
Warm the base: Return the empty pot to low heat. Add butter and let it melt.
Turn the heat off.
Build the sauce off heat: Whisk in Greek yogurt, Dijon, garlic powder, onion powder, and smoked paprika. Add 1/4 cup hot pasta water and whisk until smooth and loose.
Add the cheese: Sprinkle in cheddar (and Parmesan, if using) in small handfuls, whisking between additions. If it’s too thick, add more pasta water a splash at a time until creamy.
Season: Taste and add salt and pepper.
The cheddar and pasta water add saltiness, so adjust slowly.
Combine: Fold in the drained pasta, tossing to coat. If the sauce tightens, loosen again with a bit more pasta water until glossy and silky.
Serve: Enjoy immediately while hot and creamy. Top with a pinch of paprika or extra pepper if you like.