In a large pot of boiling salted water, cook the pasta al dente according to package instructions. Steam the broccoli on top of the pasta pot for 1-2 minutes, if possible. Drain the pasta well and blanch the broccoli.
In another large saucepan over medium-high heat, melt the butter, then add the flour to make the roux. Whisk the flour and butter until they are combined and golden, about 2 minutes.
Gradually whisk in the milk. Add the Dijon mustard and season with salt and pepper to taste. Bring the mixture to a boil, then reduce to low and simmer, occasionally whisking until sauce is thickened, about 5-7 minutes.
Stir in the cheese, then fold in the pasta and broccoli. Serve immediately
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Notes
Storage: Store any leftovers in an airtight container. They will last about 4-5 days in the fridge.Substitutes: For best results, follow the recipe as is. However here are some common substitutes that would work well in this recipe.
Use other types of shredded cheeses if you'd prefer.
Instead of milk, you can use dairy-free milk like unsweetened almond milk.
Instead of butter, you can use olive oil
To make it gluten-free, you can use almond flour or a GF 1:1 mix. Here's an excellent post on how to make gluten-free roux. You can also make this with gluten-free pasta.