Make dressing: In a food processor, whirl garlic until finely chopped. Add lemon juice, herbs, capers, anchovy, vinegar, mayonnaise, sour cream, and salt and whirl until well blended.
Make wraps: In a medium bowl, toss chicken with about 3/4 cup dressing until evenly coated.
Lay tortillas flat on a work surface. Arrange equal portions of chicken down the center of each. Evenly top chicken with bell pepper, mushrooms, cheese, and spinach, then drizzle 1 tbsp. more dressing over each wrap's filling (save the remaining dressing for another use). Fold tortillas over short sides of filling, then roll up, starting from long side of filling.
Note: Nutritional analysis is per wrap.