Preheat a grill to medium. In a medium bowl, whisk together 1/2 cup water, the mayonnaise and 1/2 teaspoon each salt and pepper; brush onto the mushroom caps, zucchini slices and chicken cutlets. Transfer the vegetables, chicken and Canadian bacon to the grill, cover and cook, turning once, until the vegetables are browned and the chicken is cooked through, 8 to 10 minutes.
Off the grill, top each mushroom cap with zucchini slices, a chicken cutlet and a Canadian bacon slice, sprinkling each layer with some of the cheese. Using a large spatula, transfer to the grill, cover and cook until the cheese is melted, about 2 minutes.