I was thinking of these too - but the ones I remember have teeny potatoes, unpeeled, which are boiled in water laced with sea salt, the water is allowed to evaporate and the potatoes are partly baked in the pan so that the sea salt forms a crust over the dried potato skins. The aioli should be fiery with fierce, pungent garlic.

We had a glut of potatoes this last week and I was scouring this thread for ideas. MrLapLap's been working hard and I couldn't get out by myself so I had to make do with what I had (couldn't get dill or chives and I'd run out of capers).
I was inspired to make a potato salad version of the Spanish version of Russian salad.

6 medium potatoes (boiled and cubed)
2 large carrots (boiled and diced)
1 large red bell pepper - roasted (Still warm)
2 large pickled gherkins (finely diced)
4 small pickled onions (finely diced)
1 drained can of tuna
Extra virgin olive oil
Good flavour mild vinegar
mayonnaise
salt & pepper

Cut red bell pepper into pieces and place in bowl with a few generous slugs of vinegar. Stir and add in tuna. Add gherkins and onions, combine well, and add carrots, olive oil and seasonings, stir again. Toss in potatoes, check for seasoning and combine with a little mayonnaise to taste.

You can finely chop a boiled egg or two and add this as well.

Capers would be a good addition if you had any.

To be honest, it's pretty similar to the Spanish 'Russian' salad, but it's a nice way of using up carrots (I usually hate them boiled and don't have an oven) and inferior pickles (we usually have Polish gherkins but tried a brand from Turkey - the vinegar was too harsh - they were perfect for this recipe)