To make the burgers: In a medium saute pan, toast cumin seeds over medium heat a few minutes, until seeds release their aroma and darken slightly. Pound seeds in a mortar or spice grinder until coarsely ground.
Return pan to the stove over high heat for 1 minute. Add olive oil and shallots. Turn heat down to medium-low, and cook for a few minutes, stirring once or twice, until shallots start to soften. Add garlic, thyme, cumin, and sliced chile. Season with 1/4 teaspoon Kosher salt and a few grindings of black pepper and cook 3 to 4 minutes, until shallots become translucent. Set aside to cool.
In a large bowl using your hands, combine ground pork, chorizo, bacon, cooled shallot mxture and parsley, being careful not to overmix the meat. Season with 1 1/4 teaspoons Kosher salt and lots of freshly ground black pepper. Shape meat into six 6-ounce patties. Chill if making in advance.
Light the grill 30 to 40 minutes before cooking, and remove pork burgers from the refrigerator to come to room temperature (if you made them in advance).
When coals are broken down, red, and glowing, brush pork burgers with olive oil, and grill them 3 to 4 minutes on the first side, until they’re nicely browned. Turn burgers over and place a piece of cheese on each one. Cook another 3 minutes or so, until pork is just cooked through. (It should still be slightly pink in the center.).
Slice buns in half, brush them with olive oil and toast them on the grill, cut side down, for a minute or two until they’re lightly browned.
Spread both sides of the buns with mayonnaise. Place a burger on the bottom half of each bun and dollop with mayo. Place some arugula or lettuce leaves on top and finish by covering with the top half of the bun.