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In a dutch oven (I have this one from my in-laws a few years ago, and I LOVE it) or stockpot, bring the broth, carrots, celery, potatoes, and onion to a boil, then reduce the heat to a simmer. Stir, and cover the pot and let cook for 10 minutes until the vegetables start to soften.
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Meanwhile, cook, in a large frying pan, cook the bacon until crisp. Drain the bacon on a paper towel-covered plate, leaving the bacon drippings in the pan.
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Add the broccoli to the large pot of vegetables, cover and let cook for another 10 minutes.
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Whisk the flour into the bacon drippings (if it's totally gross to you to use bacon drippings, wipe out the pan and use 3 Tablespoons of butter instead, but really why would you bother cleaning the pan an extra time and waste all the bacon flavor?). Let the flour and drippings cook, stirring constantly, for about 30-60 seconds. Pour in the milk, whisking all the while, and let cook until it starts to bubble and thicken (about five minutes), whisking frequently.
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Add the cheddar cheese to the milk mixture in four parts, mixing well between each addition to let it melt. Add the salt, pepper, and dried mustard, then add to the pot of broth and vegetables.
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Sir thoroughly to combine, crumble in the bacon slices and salt and pepper to taste.
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Top with chopped chives or green onions, if desired, and serve warm!