For dressing: Whisk all ingredients together in a bowl and store in a jar. Can be made 3 days ahead.
For Salad: Preheat 2 burners of your grill to medium-high heat.
Season both sides of chicken thighs with salt and pepper. Brush both sides of sourdough bread with olive oil.
Place chicken and bread onto grill directly over burners. Let chicken and bread cook for 3-4 minutes per side.
Check chicken at 4 minutes with a tong and if there is no resistance, flip the chicken. If there is resistance, let cook for another minute (to avoid meat sticking). Flip bread as well, and when it's golden brown on both sides, move to side of grill that is off.
Let chicken thighs cook for another 3-4 minutes, until chicken reaches internal temp of 160 degrees F.
Move chicken and bread to a platter and let rest for 10 minutes. Then slice chicken into thin strips, and tear grilled bread into small pieces.
Assemble the salad with romaine in the bowl, tomatoes, cucumber, avocado, parmesan, grilled chicken, and grilled bread.
Pour 1/2 of salad dressing over the top and toss to coat, just before serving. Add more dressing to top of salad if you'd like. Serve.