Heat the oven to 375°F. Toss bread cubes with 1 teaspoon olive oil on a small sheet and bake for 8 to 10 minutes until crisp, turning once; cool completely.
Heat a grill pan over medium-high heat. Rub chicken with remaining oil, salt and chicken pepper and grill for 4 to 5 minutes per side until the centers reach 165°F.
Cut chicken into short thin strips, spread in a shallow container and refrigerate promptly for about 60 minutes until 40°F or colder; chilling is included in prep time.
Whisk yogurt, mayonnaise, Parmesan, anchovy paste, garlic, lemon juice, Dijon and dressing pepper until smooth. Toss cold chicken and dry romaine with the dressing, then fold in cooled croutons just before rolling.
Divide filling below the center of each tortilla, leaving a 1-inch border. Fold the sides inward and roll tightly from the bottom, then rest seam-side down for 1 minute.
Cut across the middle and serve cold or refrigerate promptly at 40°F or below, transporting with ice packs.