Place tomatoes and garlic cloves on a rimmed baking sheet. Drizzle with 2 tablespoons olive oil and balsamic vinegar. Season generously with salt and pepper and toss to coat. Transfer to oven and roast until tomatoes are burst and caramelized, shaking pan occasionally, about 30-35 minutes. Remove from oven and let cool. Reduce temperature to 375 degrees F.
Toss bread cubes with remaining 2 tablespoons olive oil. Season with salt and pepper and transfer to oven. Toast until lightly golden on the edges, about 10 minutes. Transfer to a bowl and immediately toss with Parmigiano-Reggiano.
Combine kale, cherry tomatoes, and croutons in a large salad bowl. Toss with enough dressing to coat. Season to taste with salt and pepper and toss again. Divide among serving plates and top with additional Parmigiano-Reggiano.
For the Caesar Dressing:
Combine olive oil, vinegar, egg yolks, anchovies, garlic and 1 tablespoon water in a blender and blend until smooth. Alternatively, you can use an immersion blender and place ingredients in a glass measuring cup. Fold in Parmigiano-Reggiano and season to taste with salt and pepper. If necessary, thin with additional water, 1 teaspoon at a time, until desired consistency is reached. Leftover dressing can be kept in the refrigerator for up to one week.