Prep the chicken: Season the chicken thighs all over with salt and let them sit on a plate at room temp for 15 minutes.
Make the Caesar dressing: In a medium bowl, mash the anchovy fillets into a paste. Add the grated garlic, egg yolk, lemon juice, and a pinch of salt. Whisk until smooth. Slowly drizzle in the olive oil while whisking constantly until thick and emulsified. Stir in 1/4 cup of the grated Parmesan.
Set up the breading station: Pour the buttermilk into a shallow bowl. In another shallow bowl, combine the panko, remaining 1/4 cup grated Parmesan, a pinch of salt, and black pepper.
Bread the chicken: Working one piece at a time, dip the chicken into the buttermilk, letting excess drip off. Press into the panko mixture, coating fully on both sides. Transfer to a plate.
Fry the chicken: Heat the neutral oil in your skillet over medium-high heat until shimmering. Carefully place the breaded chicken in the pan, working in batches if needed. Cook for 3-4 minutes per side, until golden brown and cooked through. Transfer to a paper-towel-lined sheet pan and sprinkle immediately with flaky salt.
Toast the bread: Pour off excess oil, leaving about a tablespoon in the pan. Place the ciabatta rolls cut-side down in the hot pan. Toast for about 1 minute, until golden and crispy.
Assemble: Spread a generous layer of Caesar dressing on the bottom half of each roll. Add a layer of shaved Parmesan (moisture barrier). Top with sliced romaine, then the crispy chicken. Drizzle more dressing over the chicken, add another sprinkle of Parmesan and a squeeze of fresh lemon. Close the sandwich and press gently.