Melt the butter in a saucepan over a medium heat until it foams. Add the chopped potatoes and onions, season with salt and pepper, then stir well and cover with a butter wrapper or a piece of parchment paper. Add the pan lid and sweat over a gentle heat for 10 minutes, stirring occasionally to prevent the potatoes sticking.
Add the stock, bring to the boil, and cook until the vegetables are all tender. Add the chopped herbs and milk (or milk and cream), liquidize the soup and season to taste.
While the vegetables are cooking, peel the chorizo and cut into small dice. Pour the olive oil into a cool frying pan. Add the chorizo, then place the pan on a very low heat and gently cook for a few minutes, turning the chorizo every so often. Done over a very low heat like this, you’ll end up with beautifully cooked chorizo with the rich amber coloured oils rendered out. You want both the oils and the chorizo itself for drizzling over the soup when serving. Take off when it is crisp, reserving the rendered oil.
Reheat the soup if necessary, then pour into warm bowls and top with a few pieces of cooked chorizo, with a drizzle of the oil from the pan over the top.