Here is a dish which requires a scheme rather than a recipe. That's true of soups in general, and of stovetop cooking in an even broader general. But dishes like potato soup have a simple enough flavor profile that the way to make them 'right' is an open approach to adjustment.
I made a leek and potato soup for my Mom's birthday yesterday, and it turned out well! It was served with topping options: crisp bacon crumbles, extra sharp cheddar cheese, blue cheese crumbles, chopped chives, sour cream, saltine crackers. I also made rolls and prepared some buttered corn, with the idea that a person could even add their serving of corn to the soup. I did.
Here was my scheme:
Melt ~6 T butter in a very large pot
Saute ~6 leeks, halved lengthwise, carefully washed, and thinly sliced, in the butter, keeping the pot covered except for frequent stirrings. ~15 minutes.
add ~4 very large potatoes (a few pounds), peeled and diced small.
Again, cover and cook with frequent stirrings, over medium heat, ~20 minutes until potatoes are tender. Don't let them burn; I did at this point and had to transfer to a new pot.
Add ~8 c chicken stock. Bring to the boil. Reduce heat, cover, and simmer until everything is quite tender.
Add creaminess: I added a pint of half-and-half, and, wanting more creaminess, planned to add sour cream; but decided I did not have enough to add to the soup and supply as a garnish. So I got crazy, and added a heavy dollop-- almost a cup? cottage cheese as well. Or you could just add cream.
Season heavily with salt, pepper.
Take off heat; blend with immersion blender till quite smooth, adjusting seasoning.
Serve with: bacon, cheeses, chives, sour cream, crackers, corn, etc.