For the garlic mayo: Mix together the mayonnaise, chili powder, cumin, garlic and a little salt and pepper in a medium bowl; set aside.
For the tostones: Heat a medium nonstick skillet over medium heat. Add the olive oil and ham and cook, tossing occasionally, until seared, 2 to 3 minutes. Set aside.
Heat a 14-inch skillet over medium-high heat and add the neutral oil. Fry the tostones until golden brown and crispy, about 2 minutes per side. Remove to a paper towel-lined plate or baking sheet fitted with a cooling rack. Season with salt immediately.
To build the tostone bites: Arrange the tostones on a serving platter. Top each with a dollop of the garlic mayo, followed by some of the ham, lettuce, Cheddar and tomatoes.