1. Preheat oven to 375°F. Coat a 13 X 9 X 2 inch baking casserole with 1 tablespoon butter.

2. Bring a large pot of abundantly salted water to a vigorous boil, then add the broccoli stems and cook for 4 minutes. Add the macaroni and cook for 3 minutes, then add the broccoli florets and cook until the macaroni is al dente or even a little harder, about 5 minutes. Drain well, and rinse under cold running water if you think the broccoli and the macaroni do not need any more cooking, otherwise, set them aside in a colander to Drain.

3. In a saucepan, melt the remaining 3 tablespoons of butter over medium-high heat, then add the shallots and cook, stirring, until softened, about 3 minutes. Add the flour and cook, stirring, to form a roux, for about 2 minutes. Remove the saucepan from the heat and whisk in the milk and cream. Season with salt and pepper. Return the saucepan to the heat and bring the sauce to a boil, stirring. Reduce the heat to medium and continue to cook for 3 minutes. Add 2 cups of cheese and once it has melted, continue to cook, stirring, for another 2 minutes.

4. Mix the cheese sauce in a large bowl with the cooked macaroni and broccoli. Transfer to a casserole dish. Smooth the surface so it is even, sprinkle with the remaining cheese on top, then sprinkle on the breadcrumbs over and drizzle with the 2 tablespoons of melted butter on top. Bake until the top is crispy and slightly golden, about 25 minutes. Serve hot.