Heat the olive oil in a large heavy-bottomed saucepan over medium-low heat. Add the sliced leeks and a pinch of salt, stir to coat, then reduce the heat to low.

Cook the leeks, stirring occasionally, for 10 to 12 minutes until they are completely soft, pale, and just beginning to turn golden at the edges. Add the sliced garlic and cook for 1 more minute until fragrant.

Add the cubed potatoes to the pan and stir them through the softened leeks. Pour in the warm vegetable broth, add the Parmigiano rind if using, and bring to a gentle simmer.

Simmer uncovered over medium-low heat for 15 to 18 minutes until the potatoes are completely tender when pierced with a knife tip. Remove and discard the cheese rind.

For a partially smooth texture, use an immersion blender to blend about half the soup directly in the pot, leaving visible chunks of potato and leek. For a fully chunky soup, skip this step. For a smooth vellutata, blend the whole pot.

Taste and adjust salt and black pepper. Ladle into bowls, drizzle each serving with a little raw olive oil, and scatter a few thyme or parsley leaves on top. Serve immediately with toasted bread on the side.