1-First Step: Prep and mise en place (5 minutes) Chop the onion finely, peel and chop the carrots, and roughly chop the broccoli into small florets if using fresh. Measure out the butter, flour, broth, half and half, and spices so everything is ready to add as you work. Freshly shred the cheese if possible; it melts better than pre-shredded varieties.

2-Second Step: Sauté the aromatics (2-3 minutes) Heat 1 tablespoon butter in a dutch oven over medium heat. Add the chopped onion and sauté until soft and translucent, about 2 to 3 minutes. Keep the heat moderate so the onion sweetens without browning too much.

3-Third Step: Make the roux (1-2 minutes) Add 1/4 cup salted butter to the pot. When melted, whisk in 1/4 cup all-purpose flour and cook for about 1 minute, whisking often. The roux should smell slightly toasted but not browned; this step removes the raw flour taste and is the base for a silky soup.

4-Fourth Step: Add the liquids and whisk (2-3 minutes) Gradually pour in about 1 cup of chicken broth while whisking to form a smooth base. Then add the remaining chicken broth (1 more cup) and 2 cups half and half, continuing to whisk to avoid lumps. Let the mixture cook for a couple of minutes while whisking frequently; this helps the roux fully hydrate and thicken the liquid.

5-Fifth Step: Add vegetables and seasonings (10-15 minutes) Stir in 12 ounces chopped broccoli, 2 chopped carrots, 1/4 teaspoon paprika, 1/4 teaspoon garlic powder, 1/4 teaspoon dry mustard (or 1/2 teaspoon regular mustard), 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper. Reduce heat to low or medium-low and simmer for 10 to 15 minutes until the soup has thickened and the vegetables are fork-tender. Add a splash of broth or half and half if the soup becomes too thick.

6-Sixth Step: Blend to preferred texture (2-5 minutes) Blend the soup to your preferred texture using an immersion blender directly in the pot, or transfer in batches to a countertop blender. For a chunky soup, pulse briefly so you keep some florets whole. For an ultra-smooth soup, blend until silky. When using a countertop blender, let the soup cool slightly and vent the lid to avoid pressure build-up.

7-Seventh Step: Add the cheese (3-5 minutes) Remove the pot from direct heat and add 2 cups shredded cheddar cheese in handfuls, stirring until each addition melts before adding more. Adding the cheese off the heat helps prevent it from breaking or becoming grainy. Adjust seasoning with more salt and pepper if needed.

8-Final Step: Serve and garnish Serve hot in bowls and garnish with extra shredded cheddar, a crack of black pepper, and crunchy croutons if you like. A squeeze of lemon brightens the soup at the table if desired. Leftovers refrigerate well for a few days.