Arrange a rack in the middle of the oven and heat the oven to 425ºF. Coat an 8-inch square or other 2-quart baking dish with cooking spray or butter and place on a baking sheet.
Melt butter in a Dutch oven or large pot over medium heat. Transfer 1 tablespoon of the melted butter into a small, heatproof bowl.
Add yellow onion to the melted butter in the pot and cook over medium heat, stirring occasionally, until softened, about 5 minutes. Sprinkle with flour and mustard and stir constantly for 1 minute. While stirring constantly, slowly pour in the milk and stir until the flour is completely dissolved. Bring just to a full simmer over medium-high heat, stirring frequently.
Add shredded cheddar cheese and salt, and stir until the cheese is melted and the sauce is bubbling. Remove the pot from the heat, add broccoli, and stir until well-coated in the sauce. Transfer to the baking dish and spread into an even layer.
Add breadcrumbs and grated Parmesan cheese to the reserved bowl of melted butter and stir to combine. Sprinkle the gratin with the breadcrumb mixture.
Keeping the gratin on the baking dish to catch any drips, bake until the broccoli is tender and the sauce is bubbling, about 30 minutes. If you’d like a browner top, broil until the breadcrumbs are golden-brown, 2 to 3 minutes more. Let cool for 10 minutes to let the sauce thicken before serving.