Preheat a grill or griddle over medium heat.
Season the chicken breasts lightly with salt and pepper, then grill 6–7 minutes per side until an instant-read thermometer registers 165°F (74°C). Remove and let rest briefly.
While the chicken cooks, whisk together lemon juice, olive oil, Worcestershire sauce, Greek yogurt, chopped garlic, 1/2 teaspoon salt, 1/2 teaspoon black pepper, and dried oregano in a small bowl; set the dressing aside.
Dice the cooked chicken and place it in a large bowl. Pour half of the dressing over the warm chicken and toss to coat.
Add the chopped lettuce and croutons to the bowl with the chicken, pour in the remaining dressing, and toss gently to combine.
Spoon the chicken salad evenly onto the whole-grain tortillas, sprinkle with shaved or grated Parmigiano Reggiano, roll up each tortilla, and cut in half to serve.