Grill your chicken, grill your salad, and grill your croutons!
Prepare the grill for medium-high heat. In a large bowl, using a fork, mash the anchovies, garlic, and 1/2 tsp. salt into a paste. Whisk in the lemon juice and egg yolk. Gradually whisk in the grapeseed oil; season the dressing with salt and pepper.
Brush the chicken with 1 tbsp. olive oil; season. Grill until cooked through, about 4 minutes per side. Transfer to a cutting board; let cool. Slice crosswise into 1/2-inch strips.
Brush the cut sides of the romaine and the baguette slices with the remaining 4 tbsp. olive oil; season. Grill, turning occasionally, until charred in spots, 1 to 2 minutes.
Arrange the romaine, baguette slices, and chicken on a platter. Drizzle with one-third of the dressing; season. Top with the Parmesan. Serve with the remaining dressing.