In a large pot, combine the cauliflower, carrot, celery, onion, and chicken broth. Bring mixture to a boil then reduce heat; Cover and simmer for 12-15 minutes or until vegetables are tender: The kitchen will begin to smell faintly sweet and vegetal as the broth warms, with steam clouding your glasses if you wear them. You want to hear a steady gentle simmer with occasional soft bubbling, not a vigorous roll, because a violent boil can break apart the florets too quickly and make the liquid cloudy. The vegetables should yield to a fork easily when pressed, signaling they are cooked through. This stage extracts flavors into the broth and softens the vegetables for blending. One common mistake is over boiling which can cause the cauliflower to fall apart and develop a grainy texture when pureed. If the broth reduces too much, add a splash of water to maintain the desired liquid level, and keep the lid slightly ajar to control evaporation. Stir gently once or twice during simmering to ensure even cooking and to prevent sticking to the pot bottom.
In another large saucepan, melt the butter for the roux. Stir in the flour, salt and pepper until smooth. Whisk the milk in slowly. Bring to a boil over medium heat; Cook and stir the mixture for 2 minutes or until thickened. Reduce heat. Stir in the cheese until melted.: At this point you will smell the warm, toasty aroma of cooked flour, and the mixture will transition from glossy to noticeably thicker, coating the back of a spoon. When you whisk in the milk slowly the sound shifts to a soft, steady whisking rhythm, and the texture smooths as the roux hydrates. Bringing it to a gentle boil activates the thickening power of the flour. Once thickened, reduce the heat so the milk does not scorch, which would add a burnt note. Adding the cheddar cheese should be done off the highest heat to prevent separation; it will melt into a silky ribbon that enriches the sauce. A frequent error is rushing the milk addition which creates lumps, so always temper gradually while whisking. If you spot small lumps, strain the sauce through a fine sieve or keep whisking vigorously over low heat until smooth.
Puree about 2 1/2 cups of the soup in the blender and pour back into the soup. This makes the soup extra creamy.: When you transfer hot liquid to the blender, start with small batches and hold the lid with a towel to avoid steam eruptions, which can cause scalding. You will notice a velvety, whipped sound as the blender purees the vegetables, and the color will become more uniform. The aroma will intensify, with the milk and cheese notes becoming more perceptible. Return the puréed portion to the pot and stir; the texture should be luxuriously smooth but still show a few small vegetable pieces for contrast. A key why here is that partial blending preserves both creaminess and texture, preventing the soup from becoming one dimensional. Common troubleshooting: blending everything yields a soup that can feel too heavy, while not blending at all leaves it rustic but less cohesive. Aim for balance. If the soup seems too thick after blending, add a splash of broth and adjust seasoning.
Store leftovers in the fridge.: After allowing the soup to cool slightly, transfer it into airtight containers and refrigerate. You will notice the aroma mellows as it cools, and the body firms a bit in the fridge because fats solidify. When reheating, warm gently over medium low heat while stirring to prevent scorching and to reincorporate any separated fats. The flavor often deepens after a day as the ingredients meld. A common mistake is reheating too quickly on high heat, which can split the milk and change the texture, so patience is important. If the soup tightens up after chilling, add a little warm broth or milk while reheating to restore creaminess.