Preheat the oven to 200C/180C fan. Cook the pasta in boiling water for 8 minutes, adding the broccoli florets for the last 4 minutes. Drain and set aside.
While the pasta cooks, melt the butter in a saucepan over medium heat. Add the onion and cook for 3 minutes until soft.
Stir in the flour and cook for 1 minute. Gradually whisk in the milk until smooth and thickened, about 3 minutes.
Remove from heat and stir in the Stilton, mustard and black pepper until the cheese has melted.
Combine the pasta and broccoli with the sauce in an ovenproof dish. Top with the grated cheddar.
Bake for 15 minutes until golden and bubbling on top.