Adjust oven rack to center position and heat oven to 425 degrees.
Mince garlic in a food processor or blender. With motor running, slowly add 1/2 cup of the olive oil. Continue to process for about 30 seconds, then strain oil through a fine-mesh strainer. Add 2/3 of the garlic back into the oil, and reserve remaining garlic for the dressing.
Arrange bread on a wire rack set over a baking sheet. Brush garlic oil over bread. Toast until crisp and golden, about 7 minutes. Meanwhile, whisk mayonnaise, lemon zest and 2 Tbs. lemon juice, Worcestershire, grated Parmesan, and reserved garlic in a small bowl. (It can be covered and refrigerated for several hours.)
When ready to serve, toss dressing with chicken; season to taste with salt and pepper. Toss salad with remaining 2 tsps. of olive oil, and salt and pepper to taste. Drizzle in remaining 1 Tb. lemon juice; toss again. Top each toast with a portion of greens, chicken and a Parmesan shaving. (Toasts can stand uncovered for up to 2 hours.)