Get your grill going to medium-high, around 400°F, for direct heat. While it heats, pat your salmon fillets completely dry, then brush them with a little olive oil. Season with salt and pepper and set them aside while you prep everything else.
Halve your lemons crosswise and brush the cut sides with oil. Slice the sourdough into thick pieces and brush with oil and a pinch of salt. Cut the romaine hearts in half lengthwise and brush generously with oil and salt too.
Oil your grill grates well before adding anything on top. Place the lemons cut-side down, the bread slices flat, and the romaine cut-side down, and grill them together if there's room. This takes about 3 to 8 minutes depending on the item.
Reduce your grill heat slightly, then place the salmon skin-side up and cover it. Let it cook for about 3 minutes, then carefully flip with a fish spatula and cook another 3 minutes. You're looking for an internal temperature around 125°F to 145°F.
Juice your grilled lemons into the blender, then add the grated Parmesan, anchovies, garlic, mustard, and pepper. Blend until smooth, then slowly stream in the remaining olive oil while the motor runs until it turns thick and creamy.
Chop your grilled romaine and pile it into bowls as the base. Break the salmon into large chunks, removing the skin, and place it right on top. Scatter the croutons over everything and finish with a generous drizzle of that smoky Caesar dressing.