Bring a saucepan three-fourths full of water to a boil over high heat. Add 1 teaspoon salt to the boiling water and stir to dissolve. Reduce the heat to medium, add the potatoes, and cook until tender when pierced with the tip of a sharp knife, 25-30 minutes.

Drain the potatoes In a colander and return to the warm pan. Place over medium-low heat and stir to steam-dry the potatoes, about 2 minutes. Pass through a ricer or use a fork or potato masher to mash the potatoes in the saucepan until smooth. Add the eggs, parsley, 1/2 teaspoon black pepper, and the cayenne and stir to mix well. Beat in the flour. Taste and adjust the seasoning.

In a bowl, toss together the ham, green onions, and cheese. Place the bread crumbs in a deep bowl. To make the croquetas, dampen your hands with water.

Scoop up about 1/3 cup of the potato mixture with your hand and pat with your palms to flatten it Into a patty about 3 inches in diameter.

Cup your hand slightly, then place about 1/2 tablespoon of the ham mixture in the center of the patty. Gently fold up the edges of the patty to enclose the filling, and roll the croqueta between your palms to give it a round shape. Roll the croqueta in the bread crumbs to coat thoroughly on all sides and place on a plate.

Repeat to make the remaining croquetas. (If you prefer, gently flatten the croquetas into thick patties before frying.)

Pour oil to a depth of 1 inch into a heavy saucepan over medium-high heat and heat to 350°F on a deep-frying thermometer. Place a few croquetas in the hot oil, being careful not to crowd the pan, and fry, using tongs to turn as needed, until crisp and well browned on all sides, 5 minutes total.

Transfer to paper towels to drain. Repeat to fry the remaining croquetas. Serve hot, or let cool for 5-10 minutes and serve warm.