Cook the bacon slices in a large skillet over medium-high heat until crispy, about 8-10 minutes. Transfer to a paper towel-lined plate to cool, then crumble into bite-sized pieces. While the bacon cooks, prepare your mise en place: dice the potatoes into 1/2-inch cubes, cut carrots and celery into 1/4-inch pieces, finely dice the onion, and cut broccoli into 1-inch florets. Shred your cheddar cheese fresh (this melts much more smoothly than pre-shredded varieties which contain anti-caking agents).

In a large pot, combine the chicken broth, potatoes, carrots, celery, and onion. Bring to a boil over high heat, then reduce to a gentle simmer and cover. Cook for 10 minutes until the potatoes are just beginning to soften but still hold their shape. This timing is important—you want them partially cooked so they finish cooking when the broccoli is added, ensuring everything is tender at the same time.

Add the broccoli florets to the simmering soup, stir gently, and cover again. Simmer for 8-10 minutes until the broccoli is tender and the potatoes are fully cooked. I prefer my broccoli just tender with a slight bite rather than mushy, so I check it at 8 minutes and don't overcook it.

While the broccoli finishes cooking, melt the butter in a separate saucepan over medium heat. Once melted, add the flour and whisk constantly for 1-2 minutes to cook out the raw flour taste—this creates a roux that will thicken your sauce. Be careful not to let it brown; you want a pale, creamy base for the cheese sauce.

Slowly pour the milk into the roux while whisking continuously to prevent lumps from forming. Keep whisking until the mixture is smooth, then continue cooking over medium heat for 5-7 minutes, stirring frequently, until the sauce thickens enough to coat the back of a spoon. Once thickened, reduce heat to low and begin adding the shredded cheddar cheese one handful at a time, stirring after each addition until completely melted before adding more. This gradual approach prevents the cheese from clumping or becoming grainy. After all cheese is melted and smooth, stir in the salt, black pepper, and dried mustard—the mustard adds a subtle complexity that makes the cheese flavor sing without being detectably spicy.

Pour the cheese sauce from Step 5 into the pot with the cooked vegetables and broccoli, stirring gently to combine everything evenly. Heat over medium heat until the soup is bubbling gently—don't let it boil hard, which can cause the milk and cheese to separate or scorch on the bottom. Stir in the crumbled bacon from Step 1. Taste and adjust seasoning with additional salt, pepper, or a splash of milk or chicken broth if the soup is too thick for your preference. Serve hot in bowls.