Boil potatoes with the salt, in the peel, until thoroughly cooked. Peel when still warm. Grate through a fine grater. Place in a large bowl.
Make a hole (or well) in the center and drop in the eggs. With your fingers, work eggs into potatoes, only slightly.
Drizzle with the butter. Sprinkle with all of the flour and pepper. Now quickly work into a medium soft dough. Do not overwork.
Divide dough into 4 parts. Generously dust counter with flour and with a rolling pin, roll dough out to a 1/8" thickness. With a dough cutter or pairing knife, cut into 3" triangles.
Heat frying oil to 350°. Place triangles into oil and constantly spoon hot oil over them. Brown and turn to brown other side. (Note: If pillows do not puff fully, the oil is either too hot or not hot enough.) Place in one layer only onto a paper towel lined plate and sprinkle with extra salt. Serve at once with the following sauces: pineapple chutney sauce, guacamole, or tomato and cilantro salsa.