Bring a large pot of salted water to a boil. Add the chickpea pasta and cook it 1 minute less than the package says, usually 7 to 8 minutes. Drop the broccoli florets in for the last 3 minutes.

While the pasta cooks, add the cottage cheese, milk, Dijon, garlic powder, onion powder, smoked paprika, salt, and pepper to a blender. Blend on high for a full 60 seconds, scraping the sides once, until the sauce is completely smooth with no grain left.

Grate the cheddar from the block while the blender runs. Skip pre-shredded cheese here; the starch coating makes the sauce gritty.

Scoop out 0.5 cup of the pasta water and set it aside. Drain the pasta and broccoli and return them to the warm pot, off the heat.

Pour the blended sauce over the pasta and stir to coat. Set the pot over low heat and warm it for 1 to 2 minutes, stirring, just until the sauce is hot. Don't let it bubble or the cottage cheese can break.

Turn off the heat. Add the cheddar in two handfuls, stirring after each until it melts in. If the sauce is thicker than you like, add the reserved pasta water a splash at a time.

Taste and adjust the salt and pepper. Top with chives and serve hot.