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To make the filling, boil the red-skin or Yukon Gold potatoes in a medium (3- to 4-quart) saucepan in heavily salted water (you'll need about 2 teaspoons salt) to cover for about 10 minutes, or until they are fully tender. Drain.

In a heavy, large (12-inch) well-seasoned or nonstick skillet set over medium heat, combine the chorizo and white onion. Stir regularly, breaking up any lumps of sausage, for about 10 minutes, or until the onion is soft and the sausage is cooked through. If the chorizo doesn't render fat right away, you will want to drizzle a little vegetable oil into the pan. If the sausage has rendered more than a light coating of fat over the bottom of the skillet, pour out the extra.

Add the cooked potatoes to the skillet and continue to cook over medium heat, stirring regularly, for about 8 minutes, or until the potatoes begin to brown. As the mixture cooks, mash everything together a little with the back of a spoon or a spatula, scraping up any crusty bits of potato, so that it roughly holds together (it'll look a little like hash). Cover and keep warm over the lowest heat.

To make the salsa, combine the tomatillos, garlic, and chiles in a food processor. Pulse the machine until everything is finely chopped. Pit and peel the avocado, add the flesh of the avocado to the processor along with the water, and then pulse until everything is well blended (the salsa will be the consistency of a light mayonnaise). Scrape the salsa into a small bowl and taste and season with salt, usually about 1/2 teaspoon. Cover and refrigerate until serving.

To make the dough for the sopes, boil the baking potatoes in a medium pan in salted water to cover for about 25 minutes, or until they are fully tender. Drain and let cool. Press the potatoes through a ricer or medium-mesh sieve into a bowl. Scrape the potatoes into a measuring cup. Discard all but 1 cup of the potatoes, return the 1 cup to the bowl, and knead in the masa (fresh or reconstituted) and 3/4 teaspoon salt. The dough should be the consistency of soft cookie dough, just slightly sticky. Add a little water if the dough seems too dry.

To form and bake the sopes, heat a well-seasoned or nonstick griddle or heavy skillet over medium heat. Divide the dough into 18 portions, roll each into a ball, and cover with plastic to keep them from drying out. Then, one by one, form the fat little tortillas that will become the sopes. Cut two 6-inch squares of plastic (to be on the safe side, cut them from a food-storage bag; the thicker plastic works better for beginners). With your hands, gently press out a flattened ball of dough between the sheets of plastic to about 2 1/2 inches in diameter (it should be about 1/4 inch thick). Peel off the top sheet of plastic. Use your thumb and index finger to pinch up the dough into a border about 1/2 inch high around the edge to form the sope, the "little boat." Flip the sope-uncovered side down-onto the fingers of one hand, and gently peel off the second piece of plastic. Now, flip the sope over onto the hot griddle or skillet. After about a minute, when the sope has loosened itself from the cooking surface, remove it from the griddle. This cooking is just to set the bottom surface, not to cook the masa all the way through. While the first sope is cooking, continue shaping and adding others to the griddle or skillet. After cooking, to keep them from puffing oddly during frying, prick the bottom of each sope with a fork, being careful not to go all the way through. Cool, then cover the sopes with plastic to keep them from drying out.

To finish the sopes, heat 1/2 inch of oil to 350ºF in a heavy, deep medium skillet or saucepan. (It's best to use a thermometer; lacking one, you can judge the temperature by dipping the side of a sope in the oil: if it sizzles vigorously, it's ready.) Meanwhile, turn on the oven to its lowest setting and set out a tray lined with paper towels. A few at a time, fry the sopes for about 1 minute, or until they are a rich golden brown. Drain them upside down on the paper towels, then keep them warm in the oven.

When all the sopes are done, arrange them on a serving platter. Spoon about 1 1/2 tablespoons of the warm filling into each one, place a dollop of avocado-serrano salsa on top, sprinkle generously with the cheese, and top with a cilantro leaf. Serve without hesitation.

Working Ahead: You can prepare the filling a day or two ahead and reheat it, either in the microwave, in the oven covered with foil, or in a skillet over medium to medium-low heat. Add a little water if the mixture seems dry. The sope bases can be baked several hours ahead; cover them and leave at room temperature. Finish the frying just before you want to fill and serve. The avocado salsa is best made just a few hours ahead.