For the dressing: Combine the yogurt, oil, lemon juice, mustard, anchovy and garlic in a blender with 3 tablespoons of water and blend until smooth. Add the Parmesan and pulse to combine, then season with salt and pepper.
For the salad: Heat the olive oil in a large skillet over medium heat. Add the garlic and cook, stirring, until aromatic and golden, about 3 minutes. Remove the garlic. Brush the cut side of the lettuce with some of the garlicky oil.
Add the bread cubes to the skillet with the remaining oil and cook, stirring often, until golden brown on all sides, about 5 minutes. Transfer to a bowl and season lightly with salt and pepper.
Heat a large, double-burner grill pan over medium-high heat. Season the lettuce with salt and pepper. Grill the cut side of the lettuce until marked and just starting to wilt, about 3 minutes.
Serve the lettuce and croutons drizzled with dressing and topped with Parmesan. Unused dressing keeps refrigerated in an airtight container up to 1 week.