Add the egg yolks, anchovy fillets, garlic, and neutral oil to a tall container or wide-mouth jar. Blend with an immersion blender until the mixture is fully emulsified and thick.
Add the fresh lemon juice, lemon zest, and basil if using. Blend again until smooth, then season with kosher salt and freshly cracked black pepper. If making a traditional Caesar dressing, fold in the shaved Parmesan. For this sandwich, reserve the Parmesan for assembly.
Toss the chicken thighs with the neutral oil, paprika, garlic powder, citric acid, kosher salt, and black pepper until evenly coated.
Heat a cast iron skillet over high heat. Add the chicken thighs and cook undisturbed until a deep golden crust develops. Flip and cook the second side until well browned.
Transfer the skillet to a 400°F oven and cook until the chicken reaches an internal temperature of 165°F. Let the chicken rest for several minutes before chopping it into bite-sized pieces.
Slice the baguettes in half and toast the cut sides in a skillet until golden brown.
Top the toasted baguettes with the chopped chicken. Generously shave Parmesan over the chicken and allow some cheese to fall directly onto the hot skillet. Transfer the sandwiches to the 400°F oven until the Parmesan is melted, bubbly, and lightly browned with crispy edges.
While the sandwiches finish cooking, toss the Bibb lettuce with the Caesar dressing until lightly coated. Season with kosher salt and freshly cracked black pepper if needed.
Remove the sandwiches from the oven and pile the dressed Bibb lettuce on top of the chicken. Finish with a squeeze of fresh lemon juice.
Close the sandwiches, slice into portions, and serve immediately while the Parmesan is still crispy.