Heat broiler. In a small bowl, mix together panko and 1 teaspoon oil until coated. In a small jar, combine vinegar, 1 tablespoon mustard, remaining 1/4 cup oil, and pinch of sugar; shake to combine.
Season chicken with salt and pepper. On a rimmed baking sheet, make 4 stacks, sandwiching Canadian bacon and Swiss between 2 chicken cutlets. Brush tops with oil, season with salt and pepper, and broil until golden and cooked through, 7 minutes. Spread each stack with 1/2 teaspoon mustard and sprinkle with panko. Broil until panko is golden and toasted, 1 to 2 minutes.
Toss lettuces with half the dressing (reserve remaining for Cornbread-Stuffed Shrimp), season with salt and pepper, and serve alongside chicken.