BUTTER & SEASONINGS: In a large, heavy-bottomed stockpot or Dutch oven, melt 1 tablespoon butter and the olive oil over medium heat. Add the diced onion and cook for three to five minutes, stirring now and again, until the onion is soft. Toss in the minced garlic, mustard, paprika, cayenne, and pepper (I use a teaspoon, but feel free to add more). After adding the stock, add the salt and adjust the amount as needed. Cook, stirring regularly, for a further 45 to 1 minute or until fragrant.
ROUX: Add and melt the final six tablespoons of butter. When melted, mix in the flour with a whisk. Cook, whisking frequently, until the flour thickens, 2 to 4 minutes. This takes time to properly thicken your soup, so don’t rush it.
SOUP BASE: Whisk continuously while progressively adding the chicken stock. After adding all of the stock, add the half-and-half gradually while whisking continuously. Once the mixture has somewhat thickened, reduce heat to medium-low, add the bay leaf, and simmer for 15 minutes. Stir occasionally while chopping the broccoli.
BROCCOLI & CHEESE PREP: Meanwhile, shred the cheese and slice the broccoli. Add the matchstick or shredded carrots and chopped broccoli florets to the soup (cut the carrots in half if they are too long).
SIMMER SOUP: Simmer soup over low heat, stirring periodically, for 15 to 20 minutes, or until broccoli is crisp-tender (or tender, depending on your desire). Add salt to taste (I usually use around a teaspoon). If you’d like, you may use an immersion blender or puree a portion of the soup. We prefer it thick, but feel free to combine it to your liking.
CHIPS ADDED: Cut the heat off. One handful of cheese at a time, add it and stir until it melts. Once more, check the seasoning. Serve right away with a serving of bread for dipping in bread bowls or ordinary bowls!