Preheat a grill to medium. In a small bowl, mix together the mayonnaise, mustard, pickle and jalapeno. Spread the mustard mixture on all of the bread slices.
Shape the pork into four 4-inch patties, making them slightly oval to fit the bread; season with salt and pepper. Grill uncovered until the edges are no longer pink, about 3 minutes. Flip, top with half of the cheese and cook for 3 minutes more. Meanwhile, grill the canadian bacon (on a grill topper if you have one) until lightly charred, 1 to 2 minutes per side.
Transfer the patties to 4 bread slices. Top with the canadian bacon and remaining cheese; cover with the remaining bread slices, mustard side down. Lightly oil the grill. Transfer the sandwiches to the grill, cover and cook until the bottom bread slices are toasted, about 3 minutes. Brush the sandwich tops with a little EVOO, flip and grill until the second sides are toasted.