Add the chopped onion and garlic, stirring until softened and fragrant with a light golden color

Toss in the broccoli florets and stir to coat them in the buttery mixture

Pour in the vegetable broth, cover the pan, and let the broccoli steam until bright green and just tender

Stir in the butter beans and gently mix, ensuring they stay whole and creamy

Add the half-and-half, salt, pepper, and dry mustard, then bring everything to a gentle simmer

Let the mixture cook uncovered until the sauce thickens slightly and coats the beans

Reduce the heat and stir in most of the cheddar cheese until melted and smooth

Sprinkle the remaining cheese over the top and let it melt into a rich, creamy layer

Remove from heat once the sauce is glossy and thick, with a warm cheesy aroma