Heat the oven to 350 degrees. Line a sheet pan with parchment. Tear the bread into chunks in a range of sizes, 1 to 3 inches or so and add to the pan. Add about one-fourth of the Cheddar to the pan, drizzle with the olive oil and toss it all together so the cheese sticks to the bread as much as possible. Spread the pieces apart and bake, tossing halfway through, until the cheese has melted and the bread is crispy and golden brown, 15 to 20 minutes.

Meanwhile, in a medium Dutch oven or saucepan, melt the butter over medium. Add the onion and garlic and season with ½ teaspoon each salt and black pepper. Cook, stirring occasionally, until the onion is slightly softened and translucent, about 5 minutes.

Add the flour and the mustard and whisk into the onions for about 30 seconds. While still whisking, slowly pour in the milk so it is absorbed into the flour mixture (take your time here — working slowly and steadily will help to avoid clumps of flour).

Bring the sauce to a simmer, stir in the broccoli and season with salt and pepper. (The sauce might seem very thick at this point, but the broccoli will release its liquid as it cooks, thinning the sauce.) Cook, stirring occasionally, until broccoli is bright green and just tender, 7 to 9 minutes.

Stir in the remaining Cheddar and once it’s fully melted, stir in the beans. Bring to a simmer and then season with salt and pepper to taste.

To serve, ladle the broccoli mixture into bowls. Place Cheddar croutons on top and sprinkle with more black pepper, if desired.