Place the chicken thighs in a large bowl and season with salt and pepper. Add the garlic, anchovies and 3 tablespoons olive oil. Toss to combine all of the ingredients thoroughly. Cover and refrigerate for at least 30 minutes but as long as 8 hours.
Brush the lettuce with enough olive oil to lightly coat; season with salt and pepper. Brush each bread slice all over with Caesar dressing (you'll use about 1 tablespoon per slice).
Thirty minutes before you’re ready to cook, take the chicken out of the fridge and prepare a charcoal or gas grill for high heat, making sure to clean grates and lightly slick them with a kitchen rag or a brush dipped in canola oil.
When the grill is hot, place the chicken thighs on it and cook until browned and cooked through and they lift easily from the grates, about 5 minutes per side (an instant-read thermometer stuck into the thickest part should read 165ºF). Transfer to a platter. Brush the grates clean and lightly grease them with canola oil.
Grill the romaine until lightly charred, about 3 minutes per side. Transfer to the platter with the chicken. Grill the bread on both sides until toasted and lightly charred on the outside but still a little chewy in the middle, about 3 minutes per side. Transfer to the platter.
To serve, finely grate or shave the Parmesan over the lettuce wedges. Sprinkle with freshly ground black pepper. Place the lemon wedges on the platter and serve, passing the remaining dressing on the side.