Combine the potatoes, enough cold water to cover by 1 inch, and 1 teaspoon of the salt in a medium saucepan, and bring to a boil over high heat. Reduce the heat to medium-low. Cook, uncovered, until the potatoes are fork-tender, about 15 minutes. Drain and transfer to a large bowl.
Heat 1 tablespoon oil in a large heavy skillet over medium heat. Add the onions, parsley, garlic, jalapeño, the remaining 3/4 teaspoon salt, and the pepper. Cook, stirring often, until the onions soften, about 2 minutes. Add the ham and cook until heated through, about 2 minutes.
Add the ham mixture to the potatoes. Mash until blended but slightly lumpy. Set aside to cool for 15 minutes.
Add 1/2 cup of the flour and 1 egg to the potatoes and stir well with a wooden spoon to blend. Divide the mixture into 3 equal portions. Lightly dust a work surface with flour. Shape each portion into a 14-inch-long cylinder, about 1 1/2 inches in diameter, by rolling under your palms, coating it with flour to prevent sticking as you shape it. Cut each cylinder into 16 equal pieces. Place 1/2 cup of the flour in a small dish. Roll each piece in the flour into a ball about 1 1/2 inches in diameter. Put the balls on a baking sheet.
Put the remaining 1/2 cup flour in a shallow dish and season with 11/2 teaspoons of the Essence. Beat the remaining 3 eggs with 2 teaspoons of the Essence and 3 tablespoons water in a shallow bowl. Toss the bread crumbs with the remaining 1 tablespoon Essence in a shallow dish. Dredge each croquette in the seasoned flour, then in the egg wash, then in the seasoned bread crumbs, and return to the baking sheet.
Pour enough oil to come halfway up the sides of a large heavy pot or electric deep fryer and heat to 360°F. In batches, without crowding, fry the croquettes, turning to cook evenly, until golden brown, 3 to 4 minutes.
Sprinkle with some Essence, if desired, and serve hot.
Yield: 48 Croquettes