Start by melting 1 tablespoon of butter in a skillet over medium-high heat. Add the chopped onion and cook until it becomes soft and translucent, about 5 minutes. This brings out a mild sweetness that enhances the soup’s flavor. Once done, set it aside to use later.
In a large saucepan, whisk together the melted ¼ cup butter and flour over medium-low heat. Stir continuously for 3–4 minutes to cook the flour evenly. If the mixture looks too thick or dry, you can add a splash of milk—1 to 2 tablespoons—to keep it smooth.
Slowly pour in the milk while whisking constantly to avoid lumps. Once the milk is incorporated, add the chicken stock. Keep stirring as the mixture heats and thickens. This step usually takes 15 to 20 minutes. A smooth, creamy texture is the goal here.
Stir in the broccoli, carrots, celery, and the sautéed onion. Reduce the heat and let the soup simmer gently. Cook for 10 to 15 minutes, or until all the vegetables are tender. The broccoli should still have some bite—it adds a great texture contrast.
Remove the saucepan from the heat and slowly mix in the shredded cheddar cheese. Stir until it melts completely and blends into the soup. Taste and add salt and pepper if needed. Let the soup sit for a couple of minutes to thicken slightly before serving.