½ cup olive oil, plus more as needed
2 heads broccoli (about 2 pounds), separated into small florets, stems peeled and diced
2 ½ teaspoons kosher salt, more to taste
2 tablespoons unsalted butter
1 large Spanish onion, diced
5 cloves garlic, chopped
½ teaspoons black pepper, more for finishing
¼ teaspoon red pepper flakes
½ pound potatoes, peeled and thinly sliced
¼ teaspoon finely grated lemon zest
1 ½ tablespoons fresh lemon juice, more to taste
Grated Parmesan, to finish
Flaky sea salt, to finish
In a large soup pot, heat 2 tablespoons of oil over high heat. Add about a third of the broccoli, just enough so that it covers the bottom of the pan in a single layer without overcrowding. Cook broccoli without moving it for about 3 to 4 minutes, or until dark brown on 1 side only (leave the other side bright green). Transfer to a big bowl and repeat with remaining broccoli and more oil. When all the broccoli has been browned, season with 1 teaspoon salt and set aside.
Reduce heat to medium-low. Add butter and remaining 2 tablespoons of oil to pan. Add onions and garlic, black and red peppers, and 1/2 teaspoon salt. Cook onion-garlic mixture until soft and translucent, about 4 minutes. Add potato to the pot with 1 quart water and remaining 1 teaspoon salt. Bring to a simmer, cover pot and cook until potato is just tender, 10 to 15 minutes. Add broccoli, cover again and cook until tender, another 5 to 10 minutes.
Add lemon zest and roughly purée soup with an immersion or regular blender, leaving some small chunks for texture. Stir in lemon juice. Finish with grated Parmesan, a drizzle of olive oil, black pepper and flaky sea salt.