For the basil vinaigrette - Place the shallots and vinegar in a small mixing bowl and whisk to combine. Continue whisking and slowly add the olive oil and basil. Season to taste with salt and pepper. (This can be made in advance and stored in the refrigerator for up to 3 days.)
For the grilled chicken - Spray the grill grate with nonstick cooking spray and then preheat the grill to medium-high. Season the chicken breasts with salt and pepper. Grill the chicken on both sides until it is cooked through, about 4 to 6 minutes per side, depending on the thickness of the breasts. (The chicken can be grilled in advance and stored in the refrigerator for up to 3 days.)
For the green salad - Slice the chicken breasts into strips and place them in a mixing bowl. Add the tomatoes, mozzarella and half of the basil vinaigrette. Place the lettuce in a separate salad bowl and toss it with the remaining vinaigrette. Arrange the chicken, tomatoes and mozzarella mixture on top of the lettuce and serve.