Preheat your oven to 375°F (190°C). Place sourdough bread cubes onto a baking sheet, drizzle with 3 tablespoons of olive oil, and toss with a small pinch of salt.
Spread the bread cubes into a single layer and bake for 10 minutes until deeply golden and crispy. Remove from the oven and allow croutons to cool completely.
In a narrow glass or stainless steel mixing bowl, whisk together the room temperature pasteurized egg yolk, fresh lemon juice, minced garlic, anchovy paste, and dry mustard powder until well combined.
Whisking rapidly and continuously, add 1/3 cup of extra virgin olive oil a few drops at a time. As the dressing thickens and lightens in color, slowly pour in the remaining oil in a thin, continuous stream until smooth and creamy.
Place the chopped, dry romaine lettuce into a large wooden or ceramic salad bowl. Pour the emulsified dressing over the greens.
Toss gently with salad tongs until every leaf is evenly coated and glossy.
Scatter the cooled sourdough croutons, grated Parmesan cheese, and freshly cracked black pepper over the salad. Toss one final time and serve right away.