Position an oven rack in the center of the oven and preheat to 450 degrees F.
Toss the broccoli florets with the oil, 1/2 teaspoon salt and a few grinds of pepper on a baking sheet and spread out in a single layer. Roast the broccoli until browned in parts and tender but not mushy, 15 to 20 minutes. Transfer the broccoli to a cutting board to cool. Coarsely chop the broccoli.
Meanwhile, melt the butter in a large saucepan over medium heat. Add the onion and cook, stirring occasionally, until softened, 6 to 8 minutes. Add the garlic and stir until softened, about 2 minutes. Stir in the flour and cook until the flour is lightly browned and has a paste-like consistency, 3 to 5 minutes. Stir in the half-and-half and continue to cook, stirring frequently, until the mixture just comes to a bare simmer and has thickened to the consistency of heavy cream, 10 to 15 minutes. Turn off the heat and stir in the Cheddar and Monterey Jack until melted. Gently stir in the roasted broccoli and mustard. Season with salt and pepper.
Transfer to a serving dish and serve with roasted potato wedges or other dippers.
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