Preheat the oven to 400°. In a blender, puree the oil, lemon zest and juice, anchovies, and garlic. Season the dressing with salt and pepper.
On a large baking sheet lined with parchment paper, brush the chicken all over with 1 tbsp. of the dressing. In a small bowl, mix the coriander with 1/2 tsp. salt and 1/2 tsp. pepper and sprinkle all over the chicken. Bake for 10 minutes.
In a medium bowl, using your hands, massage a third of the kale (about 2 packed cups) with 2 tbsp. dressing; season. In another medium bowl, toss the bread with 3 tbsp. dressing and the Italian seasoning; season. Transfer the bread to the baking sheet and arrange the bread in an even layer around the chicken. Arrange the kale in an even layer around the bread. Bake until the bread and kale are crispy and the chicken is cooked through, 12 to 15 minutes. Let the chicken rest for about 5 minutes, then shred into bite-size pieces.
In a large bowl, whisk the remaining dressing with the grated Parmesan. Add the raw kale and massage the dressing into the kale; season. Toss in the crispy kale and bread, the chicken, tomatoes, and shaved Parmesan. Serve immediately.