Melt 1 tablespoon butter in a large skillet over medium heat.
Add ½ cup onion and the chopped garlic; cook, stirring occasionally, until softened, about 2 minutes.
Add cut broccoli, stirring until well combined.
Pour in ⅓ cup broth; cover and cook, stirring occasionally, until the broccoli is tender-crisp and bright green, 3 to 4 minutes.
Stir in rinsed beans, ¾ cup half-and-half and ½ teaspoon each pepper, dry mustard and salt.
Bring to a lively simmer over medium-high heat.
Simmer, uncovered, stirring occasionally, until the beans are warmed through and the sauce has thickened slightly, about 2 minutes.
Let cool for 2 minutes, then stir in ¾ cup Cheddar until melted.
Sprinkle with the remaining ¼ cup Cheddar.
Serve with toasted baguette slices.
