Sear the Broccoli: Heat 2 tablespoons of olive oil in a large soup pot over high heat. Add about one-third of the broccoli to the pot, arranging it in a single layer without overcrowding. Allow the broccoli to cook undisturbed until one side becomes deep brown, about 3 to 4 minutes, leaving the other side bright green. Remove the browned broccoli to a large bowl. Repeat this process with the remaining broccoli, using 2 tablespoons of oil for each batch. Once all the broccoli is browned, sprinkle it with 1 teaspoon of kosher salt.
Cook the Aromatics and Base: Reduce the heat to medium-low and add the butter along with the remaining 2 tablespoons of olive oil to the pot. Stir in the diced onion, chopped garlic, chile flakes, freshly ground black pepper, and ½ teaspoon of kosher salt. Cook the mixture, stirring occasionally, until the onions soften and turn translucent, about 4 minutes. Add the sliced potatoes, 4 cups of water, and another 1 teaspoon of kosher salt. Bring the soup to a gentle simmer, cover the pot, and cook for 10 to 15 minutes until the potatoes are tender. Once the potatoes are cooked, return the browned broccoli to the pot, cover again, and cook for another 5 to 10 minutes.
Blend and Flavor: Add the grated lemon zest to the pot. Using an immersion blender, blend the soup until it reaches your desired consistency, leaving some texture for a heartier feel. Alternatively, blend the soup in batches using a traditional blender, being careful with the hot liquid. Stir in the fresh lemon juice and taste the soup, adjusting with additional salt, pepper, or lemon juice if needed.
Serve and Garnish: Divide the soup into serving bowls and garnish each with a sprinkle of grated Parmigiano-Reggiano, a drizzle of olive oil, and a pinch of flaky sea salt. Serve the soup warm, ideally with crusty bread or biscuits on the side for dipping.