Preheat an oven to 350°F. Tear bread into 1-inch pieces into a medium bowl. Drizzle extra-virgin olive oil over the bread, and season with a big pinch of kosher salt and several turns of black pepper, tossing to coat. The bread pieces should be well-coated with olive oil. Bake until croutons are golden and crisp (15 – 20 minutes total), turning a few times while they cook. Let croutons cool before using.
Prepare an outdoor grill (charcoal or gas) to high heat. Let the grill grate preheat for 15 minutes, and clean and oil it just before cooking. Drizzle kale with olive oil, and using your hands, rub the oil all over the leaves. Sprinkle with kosher salt. Cook kale in batches on the grill over high heat , turning once, until crispy and charred at the edges (about 2 minutes total). Once cool enough to handle, cut the thick center stems from the cooked kale leaves, and chop remaining leaves. Discard stems.
To prepare the dressing, place vinegar, lemon juice, anchovy paste, garlic, and egg yolk in a wide mouth jar. Add a pinch of kosher salt and several turns of black pepper, then, using an immersion blender, blend until combined. With the blender running, slowly stream in the extra-virgin olive oil until the dressing emulsifies (thickens). If you don’t have an immersion blender, you can use a regular blender or food processor. Stir in grated Parmesan, and season to taste with additional kosher salt and black pepper.
To serve, divide chopped kale between 4 salad plates, top with torn croutons and a drizzle of Caesar dressing. Finish with a final sprinkling of grated Parmesan.