For the Chicharrónes:
In a stock pot, combine water and pork skins. Bring to a boil, turn heat down to low, and simmer for 1 hour. Let the skin cool and use a spoon to scrap off all fat. Place the skin in a dehydrator for 10 hours until it is hard as rock. (You can also dehydrate in a low temperature oven at 180°F for 6 hours.) Once dried, cut into ¼-inch pieces. Preheat a deep fryer to 400°F, and fry the skins until puffed. Season with salt and pepper.
For the Gnocchi:
Wash the potatoes under cold water to release dirt. Transfer to a medium pot and cover with cold water. Cook over low heat for 1 hour (barely reaching simmer stage), until tender. Using a towel and a butter knife, remove the skins from one potato, leaving other potato in water to stay hot. Push the potato through a tamis to process and spread milled potato to cool. Repeat with other potato and cool for 10 minutes.
Put a small pot of water on medium and bring to boil for testing gnocchi. Add salt and pepper to the spread potatoes, form into a loose ball, and make a well in the center. (Remember not to work potato too much or finished product will be gummy or tough.) Beat egg into the well and cover evenly with ½ cup flour. Mix the egg and potato together, starting in the well and working your fork outwards a little at a time.
When the mixture starts coming together, use a bench scraper and lightly fold over in a kneading fashion 6 times. Add flour as needed. (Humidity makes dough harder to work with.) On the last knead, make the shape of a ball and set to a corner of the table with a towel on top
Cut a small piece of dough the size of a tennis ball (the smaller the cut, the easier to roll). Rolling with care and a little pressure, create a log the diameter of a hot dog. Cut logs into 1-inch pieces. Add three or five gnocchi to the boiling water to test if the dough needs more egg.
Roll and cut remaining dough and keep on a sheet tray. Roll cut pieces on a textured board for added garnish.
For the Pearl Onions:
Preheat oven to 325°F. Toss the onions in olive oil and roast until tender.
For the Pistou:
In a food processor, combine the fennel, basil, garlic, cheese, salt, and pepper. Pulse to form a chunky paste. (Do not continually blend or you will have liquid.)
To Assemble and Serve:
Bring a pot of water to a boil and drop 10 pieces of Gnocchi into the pot. When the Gnocchi floats, remove from the water and place in a mixing bowl. Toss with 1 tablespoon Pistou, add the roasted Pearl Onions, and top with a couple pieces of Chicharrónes.