Special equipment: a 12-count nonstick mini donut pan and a deep-frying thermometer
Preheat the oven to 400 degrees F.
In a mixing bowl, combine the cornmeal, shortening and salt with a fork until the shortening is completely incorporated into the cornmeal. Mix in the boiling water a little at a time until a stiff dough forms, about the consistency of soft clay.
Press the dough into the donut molds in a thin even layer to completely coat each mold, including the centerpiece that would create the hole in the donut. Dip your fingers in water to keep the dough from sticking to them as you press it into the molds.
Bake until firm, 12 to 15 minutes. Let cool slightly, then carefully remove the sombreros from the molds.
Fill a medium pot with about 4 inches of oil and heat to 350 degrees F over medium heat. Working in batches, fry the sombreros until just light brown, about 2 minutes. Transfer to a baking sheet lined with paper towels and sprinkle with a bit of salt.
Fill each sombrero with salsa or guacamole.