Kitchen diary: Butter potatoes

Buttered potatoes - should they be fried or is the butter just supposed to dress the potato?

I've looked through my food diaries and my array of recipe books and depending upon the cook, the style or the country which the recipe comes from, it would seem that butter and potato have a close relationship wherever they are.

The most popular coupling seems to be frying your potato in butter.  Whether raw or cooked, the purpose of the potato is key.  If as an accompaniment to a roasted or braised meat, the potatoes seem to be raw at the outset or at least partially cooked.   This method seems to be a very northern European and Scandinavian trait. 

Fully cooked whole potatoes, albeit small potatoes generally, seem to be a mediterranean trait.  One recipe for Valencian potatoes, uses half butter/olive oil to first soften onion, garlic and tomatoes, then the cooled cooked potatoes are added with a sweet wine and then braised slowly.  The butter and wine making a relatively thick golden sauce.

Further east, cooked potato, rough cut into pieces are added to butter or gee that has been weighted with cumin and mustard seeds first along with a little turmeric and finished off, once cooked thorough, with chopped parsley.

And then there is the question about gnocchi, this lovely Italian creation was first served to me simply coated in a sage butter and, strangely, just a week later, and completely separately, new potatoes were also served in a similar way.

So, the question remains, are buttered potatoes supposed to be fried?