Bring a large pot of salted water to a boil and cook the pasta for 1 minute short of package directions, so it stays slightly firm. Drain and set aside.
While the pasta cooks, blend the cottage cheese in a food processor or high-powered blender until completely smooth, about 1 minute. Set aside—this is your secret creamy base!
In the same pot you cooked the pasta in (or a separate large saucepan), melt the butter over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly, to get rid of the raw flour taste. Slowly whisk in the milk, a little at a time, until smooth and no lumps remain. Cook, whisking often, until the mixture thickens enough to coat the back of a spoon, this should take about 3-4 minutes.
Reduce heat to low and stir in the garlic powder, mustard powder, smoked paprika, salt, and pepper. Add the blended cottage cheese and stir until fully incorporated and warmed through.
Add the cheddar and mozzarella a handful at a time, stirring until each addition melts before adding more. The sauce should be glossy, thick, and pourable, not stiff.
Fold the drained pasta into the sauce until every piece is coated. If the sauce feels too thick, loosen it with a splash of warm milk, a tablespoon at a time.
Serve immediately, with an extra dusting of smoked paprika or black pepper on top.