First, make the crunchy topping.
Preheat your oven to 425°F / 220°C. On a large baking sheet, toss together the bread crumbs, Parmesan cheese, and 1 tablespoon (15 ml / 0.5 fl oz) of melted butter until evenly coated.
Bake for 5 minutes, or until golden and crisp.
Remove from the oven, give it a stir, and set aside.
Cook the pasta and broccoli.
Bring a large pot of water to a rolling boil and add 1 tablespoon (15 ml / 0.5 fl oz) of salt.
Cook the spaghetti according to package directions. During the last 3 minutes of cooking time, add the fresh broccoli florets to the pot to cook alongside the pasta.
IMPORTANT: Before draining, use a ladle to carefully reserve 2 cups (480 ml) of the starchy pasta water.
Drain the pasta and broccoli, then return them to the hot pot.
Cover to keep warm.
Start the cheese sauce.
In a separate large saucepan, bring 1 cup (240 ml) of the reserved pasta water to a boil over medium-high heat.
Whisk in the 2 tablespoons (30 ml / 1 fl oz) of butter and the Dijon mustard until the butter is fully melted.
Melt the cheese.
Reduce the heat to low.
Add the shredded cheddar and American cheeses to the saucepan, a handful at a time.
Stir continuously and patiently after each addition, waiting for the cheese to melt completely before adding more. This creates an ultra-smooth, creamy sauce. If the sauce seems too thick, stir in a splash more of the reserved pasta water until you reach your desired consistency.
Combine and serve.
Pour the glorious cheese sauce over the warm pasta and broccoli in the pot.
Toss gently to coat everything evenly.
Serve immediately in bowls, generously sprinkling each portion with the crispy Parmesan breadcrumb mixture.
Don't forget to save the pasta water! Its starches are the secret to a silky, emulsified cheese sauce that clings perfectly to the spaghetti.
Shred your own cheese from a block. Pre-shredded cheeses contain anti-caking agents that can make your sauce grainy instead of smooth and luscious.
When melting the cheese, keep the heat low and add it in small batches. This prevents the sauce from breaking or becoming oily, ensuring a perfectly creamy consistency.