First Step: Prepare Your Ingredients
Wash and chop the broccoli into bite-sized florets. Peel and finely dice one medium yellow onion, mince three garlic cloves, and julienne or finely chop one large carrot. Have 8 ounces of sharp cheddar cheese freshly shredded and at room temperature for easier melting.

Second Step: Sauté Aromatics
In a large pot, melt 4 tablespoons of butter over medium heat. Add the chopped onion, salt, and freshly ground black pepper. Cook, stirring frequently, for about 5 minutes until the onions soften and turn translucent. Add the chopped garlic and cook for an additional minute until fragrant, being careful not to burn it.

Third Step: Make a Roux
Sprinkle ¼ cup all-purpose flour into the pot with the onion and garlic. Whisk continuously for 1 to 2 minutes until the flour turns a light golden color, creating a roux that will thicken your soup and prevent lumps. This step is crucial to achieve smooth texture.

Fourth Step: Add Liquids and Vegetables
Slowly pour in 2 cups of whole milk (or your preferred unsweetened alternative), whisking continuously to avoid lumps. Follow by adding 2 cups of vegetable broth. Stir to combine well. Add the prepared broccoli florets and chopped carrot, along with ½ teaspoon Dijon mustard. Bring the soup to a simmer, then reduce heat and cover. Let it cook gently for 15 to 20 minutes, or until the broccoli is tender but not mushy.

Fifth Step: Incorporate Cheese
Gradually add the shredded sharp cheddar cheese, stirring continuously after each handful until fully melted and the soup develops a creamy consistency. This slow addition over low heat helps prevent the cheese from clumping or becoming gloopy. Avoid boiling the soup once cheese is added.

Final Step: Season and Serve
Taste the soup and adjust salt and pepper if needed. Serve hot, optionally garnished with crunchy croutons, additional cheese, or fresh herbs for an inviting finish. For vegan or gluten-free adaptations, use plant-based milk and cheese or gluten-free thickening agents respectively, and vegetable broth to maintain dietary preferences.