In a dutch oven melt butter over medium-high heat.: Warmth will release the rich, creamy scent of the butter , and you should see it pool and shimmer before it starts to whisper at the edges. The surface should move gently when you tilt the pot. If the butter begins to brown or smoke, reduce the heat immediately because browned butter shifts the flavor and can make the base taste toasty rather than creamy. Stir the surface with a wooden spoon as it melts to distribute heat evenly, and keep a close ear for a quiet sizzle rather than an aggressive crackle.
Add in onions to dutch oven and saute until soft and translucent, about 2-3 minutes.: As the onion softens, it will release a sweet, savory perfume that layers under the butter . You should see the edges become glossy and the pieces lose their raw bite, turning almost clear. Stir every 30 seconds so they cook uniformly. If they begin to brown, lower the heat and add a splash of broth to cool the pan, because caramelized onions will push the soup into a different flavor direction.
Add in flour while whisking to form a roux, cook for about 1 minute whisking often. Continue to whisk and pour in about half of the broth.: A smooth, pale roux is the goal, and the scent will move from raw flour to a gentle toasted note. Whisk constantly so the flour and melted butter become a cohesive, paste like texture. This brief cook removes the chalky flour taste. If lumps form, keep whisking and add liquid slowly in the next step to hydrate the mixture; do not panic and dump all the liquid at once, which would trap lumps.
Add in remaining broth, as well as the half and half, slowly whisking as you pour the broth and half and half into dutch oven.: When you introduce the first portion of broth , it will loosen the roux and start to form a glossy sauce. Whisk in a steady stream to integrate without splashing. You will hear a soft hiss as hot liquid meets the roux, and the texture should turn silky. If the mixture becomes too thick, add small splashes of broth until it smooths out. Resist adding cold, large amounts quickly as that can cause separation.
Cook for about 2-3 minutes, whisking often to get out all the lumps, add in paprika, garlic powder, dry mustard, salt and pepper. Whisk to combine.: The aroma will deepen as the savory broth meets the rich half and half , and the color will become uniformly creamy. Pouring gradually while whisking prevents lumps and ensures a cohesive base. If you notice small clumps, keep whisking over low heat until they dissolve. Rapid temperature changes can cause curdling, so keep the heat moderate and steady.
Remove from heat and transfer to a blender or food processor, this is optional, blend until smooth.: The brief simmer helps thicken the base and allows the spices to bloom. You will notice the scent become fuller, with the paprika adding warmth and the dry mustard brightening the profile. Whisking will smooth any minor imperfections. If you detect any grainy texture, maintain a gentle simmer and whisk; if it persists, blending can rescue it.
Return the soup base to the dutch oven and add in broccoli florets and diced carrots.: Blending the base will create an ultra silky mouthfeel, and you will hear a steady whirl as air gets incorporated. The sound and texture change, and the aroma becomes homogenous. Be careful transferring hot liquid, vent the blender lid slightly and pulse so steam escapes to avoid pressure buildup. If you overblend, the base can become excessively aerated, which affects final thickness.
Reduce heat to medium-low and simmer for 15-20 minutes or until soup has thickened and the vegetables are fork tender, stir occasionally through cooking time.: The bright green of the broccoli and the orange flecks of carrots will contrast beautifully against the pale base. Once in, the vegetables will start to release a fresh vegetal scent. Stir them into the base to ensure even distribution. If you add them to a very hot base, they can cook unevenly, so lower the temperature slightly to allow gentle, even cooking.
Add shredded cheddar cheese a handful at a time, stirring to incorporate after each handful.: As the soup simmers, expect soft bubbling and a deepening aroma as the flavors meld. The broccoli should become tender but still retain some structure, and the soup should coat the back of a spoon when it has thickened appropriately. Stir occasionally to prevent sticking. Over simmering can make the vegetables mushy and dull the color, so test with a fork starting at 12 minutes if you prefer a bit of bite.
Season with extra salt and pepper as need and serve immediately.: Incorporating the grated cheddar cheese slowly allows it to melt into a glossy, cohesive finish without clumping. Each handful should dissolve into the warm soup, sending out a nutty, sharp cheese aroma. If the heat is too high the cheese can separate into oil and solids, so keep the pot at low heat and stir patiently. If you see oily patches, lower the temperature and whisk vigorously to bring it back together.
Season with extra salt and pepper as need and serve immediately.: Final seasoning adjusts the balance, and a small tasting spoon will tell you whether to add a touch more salt or a crack of pepper . The first bite should be warm and creamy with a clear presence of broccoli and cheddar cheese . Serve right away for the best texture, because chilled leftovers will thicken and the cheese may settle. If leftovers thicken, loosen them with a splash of broth or half and half when reheating.