Arrange a rack in the middle of the oven and heat the oven to 400℉.

Cut 6 ounces sourdough or ciabatta bread into ½-inch-thick slices. Cut or tear the bread into ½-inch pieces (about 4 cups). Place on a rimmed baking sheet. Add the oil from 1 (2-ounce) tin oil-packed anchovy fillets (about 2 tablespoons), 1 tablespoon of the olive oil, and ½ teaspoon of the kosher salt. Toss to combine, rubbing the bread with your hands so the pieces are well-coated. Arrange into an even layer.

Bake until lightly browned all over, tossing every 5 minutes, 10 to 15 minutes total. Let cool completely on the baking sheet. Meanwhile, cook the chicken and make the dressing.

Working with 1 piece at a time, place 1 ½ pounds boneless, skinless chicken breasts in a large resealable bag. Pound to an even ½-inch thickness with the flat part of a meat mallet or a rolling pin. Pat the chicken dry with paper towels and season all over with ¾ teaspoon of the kosher salt and ¼ teaspoon of the black pepper.

Heat 1 tablespoon of the olive oil in a large skillet over medium-high heat until shimmering. Swirl to coat the bottom of the pan. Working in batches if needed, add the chicken breasts in a single layer and cook until deeply browned and an instant-read thermometer inserted into the thickest part registers 165°F, 5 to 7 minutes per side. Transfer to a clean cutting board. If needed, add ½ tablespoon of the olive oil to the pan and repeat with the remaining chicken.

When cool enough to handle, cut the chicken crosswise into ½-inch-thick slices.

Prepare the following, adding each to a blender as you complete it: Juice 2 medium lemons until you have ¼ cup; finely grate 1 ounce Parmesan cheese (about ½ cup lightly packed) if needed, or measure out ⅓ cup store-bought grated; add 5 of the anchovy fillets (save the remainder for another use), 2 teaspoons Dijon mustard, 1 garlic clove, and the remaining ¾ teaspoon black pepper.

Blend until smooth, about 20 seconds. With the motor running, remove the inner cap on the lid and slowly pour in the remaining ¼ cup olive oil. Continue blending until creamy and emulsified, about 1 minute more.

Trim and coarsely chop 2 medium romaine lettuce heads (about 14 cups). Place in a large bowl and add half of the croutons and about ⅔ of the dressing. Toss until evenly coated. Top with chicken, remaining croutons, and remaining dressing. Garnish with more Parmesan cheese if desired.