Melt the butter in a large pot over medium heat. Add the chopped onion and cook until soft and translucent, about 5 minutes.: The moment the butter liquefies you will smell a warm, toasty aroma, and the pan surface should look glossy. This creates a slick surface that helps the onion cook evenly, and the gentle foaming tells you the fat is hot enough but not scorching. If the butter starts to brown quickly or smell nutty, lower the heat immediately, because burnt fat will add bitterness to the base and there is little you can do to reverse it. Use a heavy bottom pot so heat distributes evenly, preventing hotspots that can cause scorching.
Stir in the minced garlic, diced broccoli stems, and cubed potatoes. Pour in the chicken broth and water. Season with salt and pepper. Bring the mixture to a boil, then reduce heat to maintain a gentle simmer. Cook until the potatoes and broccoli stems are tender, approximately 10 minutes.: As the onion softens it becomes glossy and releases a sweet, rounded scent, which builds the soup's foundation. Stir often so the edges do not caramelize unless you intentionally want that deeper flavor. The sound will be a gentle, steady sizzle, not a violent fry. A common mistake is rushing this step on high heat, which leads to uneven cooking and sharp raw onion notes, so keep it moderate.
Using an immersion blender, purée the soup in the pot until it reaches a smooth consistency with some texture remaining. Alternatively, carefully transfer the soup in batches to a blender and purée.: Once you add the minced garlic , the aroma should lift immediately, bright and savory, and the diced broccoli stems and cubed potatoes will join to create a layered texture. Pouring in the chicken broth and water should deglaze any fond, loosening those browned bits for extra flavor. Bring to a lively boil so the surface is active, then reduce to a gentle simmer so the vegetables cook through without breaking apart. If the liquid evaporates too quickly, cover partially to maintain moisture. Overboiling can make vegetables fall apart and create a gluey finish, so be patient with the simmer.
Return the puréed soup to the pot if using a blender. Add the broccoli florets to the soup and simmer until they are tender, about 5 minutes.: You will know they are ready when a fork slides into a potato cube with little resistance and the diced broccoli stems have lost their raw snap. The aroma will shift toward a fuller, more unified vegetable scent, and the pot will sound quieter as bubbles become steadier. Avoid undercooking, which leaves a raw starchy bite, and avoid overcooking, which makes the pieces collapse and change texture drastically. Test several pieces to ensure uniform tenderness.
Remove the pot from the heat. Stir in half of the grated Parmesan cheese until well combined. Serve the soup hot, garnished with the remaining Parmesan cheese.: As you purée, the soup's color should brighten and the surface become glossy. The sound is a soft whir as air incorporates and the texture changes from chunky to velvety. Puréeing in the pot with an immersion blender is convenient and preserves heat, while a stand blender requires careful handling with venting to avoid steam pressure buildup. A typical error is over-blending until everything is completely smooth when you wanted a little texture, so stop and check frequently to maintain the balance you prefer.
Return the puréed soup to the pot if using a blender add the broccoli florets to the soup and simmer until they are tender about 5 minutes: When you reintroduce the puréed base to the pot it should look uniformly thick and coat the back of a spoon. The added broccoli florets will poach in the hot soup, turning a vivid green and offering fresh bites of texture. Simmering gently will soften them while preserving color, and the aroma becomes greener and more vegetal. Avoid vigorous boiling which can break down the florets into indistinguishable bits; a calm simmer maintains integrity and visual appeal.
Remove the pot from the heat stir in half of the grated Parmesan cheese until well combined serve the soup hot garnished with the remaining Parmesan cheese: Off the heat the temperature drops just enough that folding in half of the Parmesan cheese will melt smoothly, enriching the soup without separating. You want the texture to be silky, and the aroma will gain a toasty umami lift. Reserve the rest of the Parmesan cheese to sprinkle on each bowl for a finishing note. A frequent misstep is adding the cheese to boiling soup which can cause oils to separate, so always remove from heat first and stir gently until fully incorporated.