Preheat grill to 350°F (medium setting).
To make the dressing, combine anchovies, garlic, oil, salt, and pepper in a food processor, then process on high until smooth. Add egg and lemon juice and process until combined. Makes ¾ cup dressing.
Brush both sides of baguette slices with dressing. Place bread slices on aluminum foil and grill, turning once, 3 to 4 minutes each side, or until toasted. Cut Romaine hearts in half lengthwise, and grill with cut sides down for 2 minutes. Do not turn.
Chop Romaine crosswise into 2-inch width strips, and transfer to a medium mixing bowl. Cut the grilled baguette slices into ½ inch cubes and add to Romaine, along with the Parmigiano-Reggiano cheese. Toss and serve immediately with remaining salad dressing.