Heat a soup pot over medium-high heat. Melt the butter in the pot, then add the diced onion and sauté for 2 minutes until lightly browned and aromatic. Stir in the all-purpose flour and cook for 1 to 2 minutes.: The moment the butter melts you should notice a glossy shimmer across the pot and a gentle sizzle as the onion hits the surface, releasing a sweet, savory aroma that signals caramelization. As the onion softens its edges will become translucent and just beginning to color, which creates a flavorful base; listen for a steady gentle hiss, not a violent pop, as that indicates correct heat. When you stir in the all-purpose flour , it should coat the softened onion evenly and begin to smell toasty within a minute or two, which removes the raw flour taste and builds the soup’s body. Use a wooden spoon and keep the mixture moving to avoid forming lumps or allowing the flour to scorch, because a burned roux will introduce bitterness that is hard to fix. If your pan starts to smoke, lower the heat immediately and scrape the bottom to release any sticking flour, then proceed at a gentler temperature to maintain a clean, nutty aroma.
Whisk in the heavy cream and chicken broth. Stir the soup constantly with a wooden spoon to prevent the flour from sticking to the bottom.: As you pour in the heavy cream , you will see the pale liquid stream into the roux and begin to thicken the mixture; whisking briskly encourages a smooth emulsion so the soup does not separate. When the chicken broth joins the pot the texture becomes more fluid, and you should notice a developing scent of savory warmth as the liquids marry with the toasted flour. Keep stirring so the starch disperses evenly; this prevents a grainy or lumpy texture and ensures the soup will coat the back of a spoon. The sound changes to a steady soft simmer rather than a pop and hiss, and visually the mixture should be free of clumps. If you see small pockets of unmixed flour, increase whisking speed and scrape the pot bottom to integrate everything, because those pockets will cook into unpleasant lumps otherwise.
Add the broccoli and carrots, then bring the soup to a simmer. Continue cooking over low heat for 15 to 20 minutes, until the broccoli and carrots are softened and cooked through.: As you introduce the broccoli and carrot the aroma brightens with vegetal notes, and you will spot vibrant green florets bobbing in the creamy liquid while the orange julienned pieces soften. Bring the pot up to a gentle simmer where small bubbles rise without a vigorous boil, because that gentle cooking preserves the creaminess and prevents splitting. Over the next minutes the vegetables will give off a faint sweet scent, and you can test for readiness by piercing a floret with the tip of a knife for tender resistance. The low simmer extracts flavor into the broth and ensures the vegetables are cooked through but not falling apart. Avoid rapid boiling or the soup may reduce too quickly, concentrating salt and risking separation of the dairy; if you notice vigorous bubbling, lower heat and stir occasionally while monitoring texture.
Season the soup with salt and ground black pepper. Blend half of the soup until smooth, then return it to the pot. Add the cheddar cheese and stir until it melts.: When you season with salt and ground black pepper you should taste brightness and a warming undertone; add small increments and taste because the chicken broth already contributes sodium. Blending half of the soup creates a velvety body while leaving tender pieces of broccoli , which produces a pleasing contrast in mouthfeel, and when you swirl the blender you will hear a steady whir that transforms chunky bits into a lush base. Returning the blended portion to the pot reintegrates the texture and deepens flavor. As you fold in the shredded cheddar cheese the soup will become glossy and stringy in places before smoothing into a rich, cheesy finish; stir gently over low heat so the cheese melts evenly. A common pitfall is overheating when the cheese is added, which causes it to clump; keep the heat low and stir constantly to prevent grainy patches.
Serve immediately, garnished with additional cheddar cheese and a sprinkle of paprika on top, if desired.: When ladling the soup you should notice a warm, fragrant steam rising and the surface shimmering with melted cheddar cheese . The garnish of extra shredded cheddar cheese adds a fresh cheesy top layer, and a touch of paprika lends a subtle color contrast and mild spice. Serve while hot so the texture remains luscious and the cheese stays melty; if it sits too long the soup can thicken and lose some of its velvety quality. If you must hold it, keep the pot over the lowest flame and stir occasionally to maintain consistency, because reheating aggressively can alter the texture and make the dairy separate.