Prep the vegetables as listed. Heat 2 tbsp olive oil in a 5-quart / 4.7 L heavy-bottomed pot over medium-low heat. Add the onion, celery and carrot with a pinch of salt and cook for 8 minutes, stirring often, until the onion is translucent and nothing has browned.
Stir in the potatoes and zucchini and cook for 2 minutes. Add the water, the remaining salt and the Parmigiano rind if using. Bring to a boil over high heat, about 5 minutes, then lower to a gentle simmer, partly cover, and cook for 25 minutes. Add the frozen peas for the last 5 minutes. The potato is ready when it breaks apart against the side of the pot.
Remove the rind. Blend with an immersion blender for about 60 seconds until smooth, or pass the soup through a food mill with the fine disk. In a countertop blender, work in two batches, fill it only halfway and vent the lid. Longer blending turns the potato gluey. If the soup is too thick, stir in hot water 50 ml at a time until it ribbons off the spoon.
Return the pot to low heat and taste, adding salt if needed. Ladle into warm bowls, drizzle with the remaining 1 tbsp olive oil, then top with the grated Parmigiano, torn basil and black pepper.