Lightly crush garlic, combine in a cup with 1/4 cup olive oil and let stand for 5 minutes.
Pour garlic oil in a large salad bowl. Add red wine vinegar, lemon juice, mustard and 1/4 tsp. each salt and pepper. Reserve.
Lightly brush one side of each bread slice with some of remaining olive oil.
Heat a grill pan over medium heat. Brush chicken with remaining oil, season lightly with 1/4 tsp. each salt and pepper, add to pan and cook until golden brown and cooked through, for about 10 minutes, turning once. Transfer chicken to a board and let stand. Meanwhile, toast bread. Arrange toast on 4 plates.
Whisk dressing mixture again, add lettuce and some of Parmesan cheese and toss well. Serve salad on plates with toast. Thinly slice chicken and arrange on salads. Top with more cheese.