Peel the potatoes and slice them in chunks. Place them in a pot of cold water with 1 teaspoon salt. Place on medium heat and bring to a boil. Boil until soft (about 10 to 15 minutes after the water has begun to boil).
Remove from heat and pour the potatoes into a colander to completely drain all the water. Shake off any excess water. Put the potatoes back in the warm pot. The warmth of the pot will help to evaporate extra water which you don't want.
Use a potato masher to mash the potatoes in the pot. Add in the butter and continue mashing to get a smooth puree.
Allow the mixture to cool for several minutes then mash and mix in the egg yolks and almond meal. Mix in 1/2 teaspoon salt and 1/8 teaspoon pepper.
Cover the potatoes and place in the refrigerator to chill for at least 30 minutes.
When the mixture is cold, remove from refrigerator and shape into balls or patties or try making little pear shapes. Place the balls in the refrigerator while you are preparing the breading plates in the next step.
Prepare three breading bowls or plates. In one, place the flour. In the second, beat the two eggs with 1 tablespoon of water. In the third, stir together the almonds and bread crumbs.
Roll the balls in the flour and shake off excess flour. Then roll them in the beaten egg and finally in the bread crumb and almond mixture. Place the breaded balls back in the refrigerator for at least 30 minutes.
When you are ready to fry, heat the vegetable oil to 350-400°F (180-200°C). Lower the balls into the hot oil and cook until golden brown - about 5 minutes. Remove from oil and drain on paper towel.
Sprinkle the balls with finishing salt and serve immediately.