In a small skillet, add 1 teaspoon olive oil. When hot, add the bread and fry on both sides. Set aside.
In a pot with hot water, add the peppers and let soften for about 20 minutes.
Meanwhile, in a mortar, pound the garlic, salt, and cumin seeds, then add the pimentón and mix until a paste forms. Add the vinegar and mix.
Scrape the flesh from the peppers and add to the mortar, discarding the skins, seeds and stems. Pound the peppers and incorporate.
Break the slices of toasted bread and add to the mortar. Pound everything with the pestle until all ingredients are incorporated.
Add the olive oil in a very slow stream, stirring to incorporate to the mix. Adjust for salt.
Alternatively, transfer the mix to a blender right before adding the oil. Blend the mixture at low speed while adding the oil in a very slow stream.
Wash the potatoes and place them, skin on, in a pot. Add water until it covers them. Add the salt and boil the potatoes until tender, about 20 minutes.
Pour off the water and return the pan with the potatoes to the stove so the steam evaporates until a layer of salt has formed on the dry skins. Serve promptly.
Notes
The sauce can be stored in the refrigerator, in a covered jar, for several weeks.