In a medium saucepan whisk the pineapple juice, chili sauce, vinegar, and soy sauce. Bring the mixture to a boil, reduce the heat, and simmer until 1/2 cup remains- 15 to 20 minutes. Transfer to a medium bowl. Whisk in the mustard and sesame oil and allow to cool to room temperature.
In a medium skillet over medium heat, cook the almonds until lightly toasted, about 5 minutes, stirring occasionally.
Prepare the grill for direct cooking over medium heat (350° to 450°F/ 180° to 230°C).
Lightly brush both sides of the chicken with oil and season evenly with the marjoram, salt, and pepper. Brush the cooking grates clean. Grill the chicken breasts over direct medium heat, with the lid closed as much as possible, until the meat is firm to the touch and no longer pink in the center, 8 to 12 minutes, turning once or twice. Remove from the grill and, when cool enough to handle, cut into bite-sized pieces.
In a large bowl combine the greens, chicken, almonds, and cucumber. Drizzle the dressing over the top (you may not need all of the dressing) and toss to coat evenly. Serve immediately.