For the croutons: Preheat the oven to 350 degrees F.
Toss the bread cubes with the olive oil, Italian seasoning and a good pinch of salt and pepper. Spread the cubes out in an even layer on a baking sheet. Bake until toasted and golden brown, about 10 minutes.
For the dressing: Add the Parmesan, mayonnaise, mustard, anchovy paste, garlic, lemon juice and some salt and pepper to a food processor. Process until finely minced, 30 seconds. With the motor running, drizzle in the olive oil in a steady stream and process until emulsified.
For the salad: Put the romaine in a large bowl and drizzle with enough dressing to coat generously. Toss to combine. Add the croutons and toss lightly. Serve immediately.