Preheat oven to 450 degrees, with rack in upper third. In a food processor, pulse bread and 1 tablespoon oil until large crumbs form; season with salt and pepper. Set aside.
In a large saucepan, heat remaining 2 tablespoons oil over medium. Whisk in flour; cook, stirring constantly, 1 minute. Gradually whisk in milk. Bring to a boil; add broccoli. Reduce to a simmer. Cook, stirring occasionally, until broccoli is tender, 15 to 20 minutes. Remove from heat; stir in 3/4 cup cheese.
Transfer mixture to a shallow 1 1/2 quart baking dish; top with breadcrumbs and remaining cheese. Bake until breadcrumbs are golden, 6 to 8 minutes. Let rest 5 minutes before serving.