Add the pancetta to a cold skillet or sauté pan and set over medium heat. Cook until the fat has rendered and the pancetta is browned and crisp.
Reserve 1 tablespoon of the rendered fat and transfer the pancetta to a paper towel-lined plate.
Add the reserved pancetta fat and 1 tablespoon of olive oil to a Dutch oven over medium heat. Add the onion with a pinch of salt and cook for 3 to 5 minutes, until softened but not browned. Add the garlic and cook for another minute, until fragrant.
Meanwhile, halve the zucchini lengthwise, then slice crosswise into ¼-inch-thick half-moons. Cut the potatoes into 1-inch cubes, keeping the skin on.
Add the potatoes, zucchini, basil sprig, and 1 teaspoon of salt to the Dutch oven. Pour in enough chicken stock to just cover the vegetables, about 4 cups.
Cover and bring to a boil, then reduce the heat to maintain a gentle simmer. Cook, covered, for about 15 minutes, until the potatoes are tender and the zucchini is very soft. Add additional chicken stock as needed until the soup is the consistency you prefer.
Remove the basil stem and any large pieces of basil you can easily retrieve. Taste and season with additional salt as needed.
Ladle into bowls and top with the crispy pancetta, fresh basil, and a drizzle of olive oil.