Preheat the oven to 350 degrees F. Spread the outside of each slice of bread with ½ tablespoon of butter per side. Place one slice of bread butter side down, and spread 1 tablespoon of aïoli on the inside per slice. Divide the cheese and place half the cheese on top of one slice of bread, butter side down. Follow with a layer of chorizo, piquillo (optional) and paprika (optional). Top with remaining half of cheese. Cover the sandwich with the second slice of bread with aïoli on the inside.
Heat a 12-inch, nonstick, ovenproof sauté pan over medium heat. Add the sandwich and cook until golden-brown and the cheese has melted, 3 to 4 minutes per side, pressing down with a spatula to flatten the sandwich and integrate the ingredients. (If the bread looks golden and done but the cheese hasn't completely melted, arrange the pieces in a single layer on a cookie sheet and finish the sandwiches in an oven preheated to 350 degrees F.) Transfer sandwich to a cutting board and cut it in half, accompany with fresh potato chips, pickled vegetables and enjoy!
For the aïoli, put the yolks, lemon juice, vinegar, garlic, and salt in the bowl of a food processor fitted with the steel blade. With the motor running, slowly add the oil in a thin steam to form a thick emulsion. Transfer to a container. Cover and refrigerate the aïoli for up to 1 week. (The reason: My aïoli balsamic vinegar: Lemon juice can sometimes register as excessively acidic in aïoli. Adding balsamic vinegar smoothes out the flavor, and softens the character of the olive oil.)