Cut the broccoli into florets (about 10 cups). Trim the dry ends off the stems, peel the outer layer, and cut into coins.

Bring a medium to large pot of water to a boil. Add a generous pinch of salt, then add the broccoli stems and cook until slightly tender, about 3 minutes. Add the florets and cook until bright green and just tender, about 3 to 4 minutes. Drain and rinse with cold water for about 30 seconds to stop the cooking.

Preheat the oven to 400°F. Butter a 9x13-inch glass casserole dish.

In the same pot over medium heat, melt 1 tablespoon of butter. Add the breadcrumbs and toss to coat. Cook while stirring often until golden brown, about 2 to 3 minutes. Scrape into a bowl and set aside.

Wipe out the pot and return it to the stove. Add the remaining 2 tablespoons of butter and melt over medium-low heat. Add the flour and whisk until a smooth paste forms. Cook until the roux takes on some color and smells nutty, about 1 minute.

Slowly whisk in the milk and bring to a simmer. Whisk to ensure there are no lumps, then stir often until the mixture has thickened, about 2 to 3 minutes. Remove from heat and stir in 1 cup of cheese until smooth.

Add the drained broccoli to the casserole dish, patting it into an even layer. Season with salt and pepper, then drizzle the cheese sauce on top. Scatter the remaining cheese and top with the toasted breadcrumbs.

Bake until the cheese is melted and the top is golden brown, about 20 minutes. Let sit for 5 to 10 minutes before serving.