Preheat the oven to 350 degrees F.
For the dressing: Combine the lemon juice, anchovies, mustard, garlic, Worcestershire sauce, 1/2 teaspoon salt and a generous grind of black pepper in a mini-food processor and process until smooth. With the machine running, add the olive oil in a slow, steady stream to make a smooth, slightly thick dressing.
For the croutons: Heat the olive oil in a small skillet over low heat. Add the garlic and cook, stirring occasionally, until it is golden and begins to soften, 5 to 6 minutes. Let cool, then remove and discard garlic.
Put the focaccia in a large bowl and drizzle with the garlic oil. Season with 1/2 teaspoon salt and several grinds of black pepper and toss well to coat all of the cubes in oil. Spread on a baking sheet and toast, tossing halfway through, until the croutons are lightly golden and crisp, about 12 minutes. Let cool.
To assemble: When ready to serve, put the romaine in a large serving bowl and sprinkle with the grated Parmigiano-Reggiano. Add the croutons, drizzle with the dressing and toss well. Season with salt and pepper, if necessary. Use a vegetable peeler to shave shards of Parmigiano-Reggiano over the salad, toss gently again, and serve immediately.