Cut the florets from the broccoli stalks; cut the florets into pieces of about 1/2 inch.
Trim the rough bottoms from the stalks, peel the stalks, and cut them into 1/2-inch dice.
Heat the oil in a large pot over medium heat; add in the potatoes and cook, stirring, until they are golden and begin to stick to the bottom of the pot (don't worry if they stick; the little brown bits will add lots of flavor to the soup).
Whack the garlic with the flat side of a knife, stir it into the pot and cook for a few minutes, stirring occasionally, until you can smell the garlic.
Pour in the hot water and bring to a vigorous boil, stirring as you do to scrape up any browned bits of potato.
Toss in the bay leaves and cook for 15 minutes.
Stir in the lentils, decrease heat to simmering, and let simmer 15 minutes.
Stir in the broccoli; season the soup with salt and pepper; cook until the broccoli and lentils are very tender, about 15 minutes.
Skim any foam that rises to the surface as the soup simmers.
Remove the soup from the heat; pluck out the bay leaves, and let the soup rest off the heat, 10-15 minutes.
Ladle the soup into warm bowls and drizzle with olive oil; pass the grated Pecorino separately.