In a 5-6 quart Dutch oven or other heavy-bottomed saucepan, heat the olive oil over medium heat. Add the pancetta and cook until the fat is rendered, about 3 minutes.

Add the garlic and potatoes and cook, stirring occasionally, until potatoes begin to stick to the bottom of the pan, about 5 minutes.

Push potatoes to the edges to make an opening in the center of the pan, and add the tomato paste. Cook and stir the tomato paste in the center until it darkens a shade or two, about 2 minutes, then stir with potatoes until potatoes are evenly coated.

Add 2 quarts of water and bring to a gentle simmer. Cover and continue to simmer for about 10 minutes or until potatoes are just barely fork tender.

Add the broccoli florets and stems along with scallions and salt. Simmer, uncovered this time, until the broccoli stems are tender, the florets have broken down, and the soup is thick and flavorful, about 25 to 30 minutes more.

Meanwhile, if you want to fry the potato skins, fill a saucepan with an inch of vegetable oil and set over medium heat. When oil reaches 350ºF, add the first batch of potato skins (work in batches as you don't want to overcrowd the pan). Fry for about 5 minutes or until golden brown and crispy, then scoop out with a slotted spoon or spider and transfer to a paper towel-lined baking sheet. Sprinkle with fine sea salt while still warm. Repeat with remaining potato skins.

Spoon soup into serving bowls and top with grated cheese and fried potato skins if desired.