In a small mixing bowl, stir together the mayonnaise, ketchup, mustard, garlic, and Adobo until combined. Store the spread, covered, in the refrigerator until used.
Cook and Mash the Plantains
Soak the sliced plantains in a large bowl for 10 minutes of salted water(6 cups of water with 1/4 cup of salt) for 10 minutes to season them.
While the plantains are soaking, heat the frying oil in a 5qt dutch oven to 350°F. Once the oil is near the proper temperature, drain the soaked plantains before blotting them dry with a clean towel. When the plantains are bone-dry, cover them with a towel to keep them from being exposed to too much air while you begin frying them.
Add 15-20 slices of plantains to the hot oil and fry them for 4-5 minutes. When done, a knife should pass through the fried plantains easily and meet no resistance.
After the plantains are fried, use a spider to strain most of the oil from them before transferring them to a bowl. Don't worry about removing excess oil from the plantain pieces. The oil, along with any stock you may use, will help keep the mofongo moist.Continue to fry the plantains until they are all cooked.
Add 8 chunks of plantain to a pilón (or a mortar and pestle), followed by a handful (about 1/2 cup) of pork rinds, 1 teaspoon of garlic paste, and 1/2 teaspoon of oregano leaves. Mash the plantains until they resemble mashed potatoes. Add 8 more chunks of plantains and mash again. The mixture will look like a bowl of mashed potatoes with flecks of pork rind throughout.
You will need to mash the plantains in stages, unless you have an obscenely large pilón. After you mash your plantains and created the mofongo, squeeze a handful of the mix together. If it feels pasty and holds together, you can begin forming the the buns. If the mofongo feels dry and crumbles apart after you release pressure, add a tablespoon of stock and remix to encourage it to hold together.
Prepare the Mofongo Buns
Use a large spoon to scoop half of the mixture out of the pilón and into the center of a lightly-greased round cookie cutter. Create the bottom "bun" by firmly pressing the mofongo into the mold, make this half very flat. Carefully press the mofongo onto a lightly greased sheet tray.
Spoon the remaining mofongo into the form. Press the sides down slightly to mimic the dome of a regular bun. This is a good way to differentiate the tops from the bottom. Continue mashing and forming the mofongo buns until you have four tops and four bottoms. Cover the pan lightly with foil and transfer the pan to a warm (170°F) oven while you grill the burgers.
Grill and Assemble the Mofongo Burgers
Heat a charcoal grill using your preferred method to medium-high heat (or a gas grill to 450°F).
Press the burgers flat, leaving a dimple in the center to ensure the burgers grill as flat as possible. Season both sides of the patties with a generous pinch of adobo.
Grill the burgers for 2 1/2 minutes, then use a spatula to flip them over and grill for an additional 2 1/2 minutes: for a total of 5 minutes (for medium-rare burgers). Grill 2-3 minutes longer for medium-well or well-done burgers. Remove the burgers from the grill and allow them to rest for 5 minutes.
Remove the pan of mofongo buns from the oven and remove them from the pan using a spatula. Layer the cheese, lettuce, tomatoes, purple onions, and slather on a generous amount of burger spread on the inside of the top mofongo bun.
Place the rested burgers on the prepared mofongo buns and you are ready to serve! To avoid a big mess when eating, I suggest cutting the mofongo burgers into quarters, or just eat them with a knife and fork.