Light a grill. In a blender or mini food processor, blend the mayonnaise with the vinegar, garlic, lemon juice, Dijon mustard, Worcestershire sauce and Tabasco. With the machine on, gradually add 1/2 cup of the vegetable oil until the Caesar dressing is creamy. Season with salt and pepper, cover and refrigerate.
Lightly brush the hearts of romaine with 1 tablespoon of the oil and season with salt and pepper. Grill over moderately high heat, turning once or twice, until lightly charred but still crisp, about 3 minutes.
Lightly brush the bread with the remaining 1 tablespoon of oil. Grill, turning once, until toasted, about 2 minutes.
Spread half of the Caesar dressing on a large platter. Arrange the grilled hearts of romaine on top and brush with the remaining dressing. Garnish with the cheese shavings and serve the grilled bread alongside.