For the balsamic vinaigrette: Whisk together the olive oil, vinegar and salt and pepper to taste in a small bowl. Measure out 1/2 cup for the chicken and set aside; refrigerate the remaining vinaigrette for another use.
For the chicken and wrap: Preheat a grill or grill pan.
Place the chicken breast in a large plastic storage bag. Seal the bag, leaving a small opening, then use a rolling pin to pound the breast to a uniform thickness. Pour in about a quarter of the reserved balsamic vinaigrette, then seal the bag and toss it around to fully coat the chicken.
Grill the chicken until it's cooked in the middle, about 5 minutes per side. Set aside to cool slightly. Dice up the chicken when it's cool enough to handle.
Add the salad greens to a large bowl. Add half of the remaining dressing and toss it to coat the greens. Add more if it needs it. Add the chicken, pecans, strawberries and green onions. Toss it gently a few times until it's all combined. Top with the crumbled goat cheese and toss a couple of times.
Place an equal amount of salad down the middle of each tortilla. Roll up, then slice in half.