Whisk 1/3 cup butter in a large pot over medium heat, until melted.: Warmth brings out a comforting, nutty aroma as the butter melts and warms the pot, and you will notice a faint sizzling as it hits the metal. This initial step forms the flavor base and ensures even cooking when the flour is added. Work at medium heat so the butter melts smoothly without browning, and use a whisk to keep the fat moving. A common error is turning the heat too high, which can scorch the butter and give a bitter note, so watch the color closely.
Whisk in flour and cook for a few minutes to create a roux.: As you stir in the flour , the mixture will thicken and take on a faintly toasted scent, signaling the raw edge is gone. Continue whisking until small bubbles form and the color shifts to a pale golden hue, this tells you the flour has cooked enough to thicken without tasting raw. The texture should be smooth and ribbon like when you lift the whisk, avoid leaving dry pockets of flour. If lumps start to form, reduce heat and whisk vigorously, because under whisking will produce a grainy soup later.
Pour half and half into the flour mixture gradually, making sure to whisk as you pour. Add chicken stock to the mixture. Bring to a simmer and cook for about 20 minutes, or until thickened.: When you slowly add the half and half while whisking, you will see the roux relax and the mixture become velvety, with faint steam rising as it warms. Incorporating the chicken stock here adds savory depth and thins the base to a spoonable consistency. Bring the pot to a gentle simmer, where small bubbles form around the edges, and maintain that warmth for about 20 minutes until the body feels cohesive and slightly reduced. Stir occasionally to prevent sticking and taste for seasoning. A frequent slip is boiling too hard, which can separate the dairy and leave a curdled texture, so keep the heat moderated.
Add broccoli and carrots and simmer until the vegetables are tender (which takes an additional 20 minutes or so).: Once the base is gently simmering, add the chopped broccoli and matchstick carrot , and you will see the broth envelop the vegetables, softening their edges. As they cook, the kitchen fills with an earthy, sweet scent, and the vegetables will go from bright and crisp to softer and more opaque. Check tenderness with a fork, aiming for pieces that hold shape but yield easily, which helps the soup keep texture without becoming mush. Overcooking will turn the vegetables past tender into a lifeless mush, so taste sooner rather than later to prevent that.
Stir shredded cheddar cheese into the vegetable mixture and stir until the cheese melts.: With the pot off or on very low heat, sprinkle in the shredded cheddar cheese and you will watch it melt into ribbons, thickening the soup and releasing that tangy aroma. Stir gently to integrate the cheese evenly, creating a smooth, glossy finish. If the heat is too high, the cheese can clump and become grainy, so patience and low warmth are crucial. If you see oily separation, lower the heat and whisk slowly to help the emulsion come back together.
Season with garlic salt and pepper to taste.: At the finish, adding garlic salt and freshly ground pepper brings everything into balance and brightens the bowl. Taste in small spoons and add a little at a time, because the chicken stock and cheddar cheese will already contribute saltiness. The aroma of cracked pepper will lift the soup, but too much can mask the delicate vegetable notes, so adjust patiently. A common oversight is seasoning before the cheese is incorporated, which can lead to over salting, so always season after the final mix.
