Preheat a grill pan over medium-high heat. Put the chicken breasts in a medium bowl. Sprinkle with 3/4 teaspoon salt, a few grinds of pepper, and a drizzle of olive oil; toss to coat. Let sit at room temperature for 15 minutes.
Meanwhile, preheat the oven to 375 degrees F. Line a baking sheet with parchment and arrange the bacon strips on top. Bake, flipping once halfway through, until the bacon is golden brown and crisp, about 25 minutes. Transfer to a paper-towel lined plate.
Grill the chicken until an instant-read thermometer inserted in the middle of the chicken reads 165 degrees F, 6 to 8 minutes per side. Let cool, then slice thinly across the grain.
Slice the sourdough loaf in half horizontally and lay the two halves cut-side up. Scoop out some of the bread from both halves and reserve for another use.
Whisk the mayonnaise, Parmesan, anchovies, garlic, lemon zest and juice, 1/4 teaspoon salt, a few grinds of black pepper and the remaining 2 tablespoons olive oil together in a small bowl. Remove 1/4 cup of the dressing and add it to a large bowl along with the kale. Toss to coat and season to taste with more salt and pepper.
Spread the remaining salad dressing on both halves of the bread. Building on the bottom half, layer half the kale salad, bacon, sliced chicken, and a sprinkle of salt. Place the radishes on top of the chicken in a single layer, then add the cucumber in a single layer. Scoop the remaining kale salad into the top half of the bread, and place on top of the sandwich. Wrap tightly in parchment, wax paper or plastic wrap and secure with twine if necessary. Place a brick (see Cook's Note) or cast-iron skillet on top for at least 3 hours to flatten, refrigerated. About 45 minutes before serving, bring the sandwich to room temperature.
To save time, pick up grilled chicken breasts, which are available at most supermarket deli counters. A brick that weighs about 4 pounds works best for this sandwich. Wrap it in foil to protect kitchen surfaces, if desired.
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