Heat the oil in a Dutch oven to 375 degrees F. Fry 4 pieces of plantains until they float to the surface, about 2 minutes. Drain on a paper towel and repeat with the remaining 4 pieces. Using 2 wooden cutting boards or a heavy pan, flatten each piece of plantain. Refry each piece of flattened plantain, in batches, until golden, about 4 minutes. Season with salt and pepper and a little granulated garlic. Set aside on a paper towel.
For the sandwich build: On the bottom of 4 fried plantains (or tostones), place a slice of American cheese, a quarter of the ham, a quarter of the tomatoes and 1 lettuce leaf. Spread the Garlic Mayo on the top tostones and close. Cut in half on the bias. Pick up gently but bite aggressively.
In a bowl, mix together the mayo, chili powder, cumin, garlic and a little salt and pepper to taste.