Caesar dressing: first, you want to drain your anchovies and put them on a cutting board with your peeled garlic cloves. Start chopping them both together and get them to a fine dice. At this point, you should add a small pinch of kosher salt and mash/chop all three ingredients until you are left with a smooth paste. This may take a few minutes. You can use the side of your knife and drag it through the paste to smash and help smooth everything out. Adding the salt will help this process.
In a medium mixing bowl, add your egg yolks, lemon juice, mustard, and Worcestershire sauce and whisk to combine well.
Add the anchovy and garlic mixture to the mixing bowl with your egg yolk mixture and whisk again for a minute to make sure everything is incorporated.
The next part is the magic in the dressing-making process. This is the point where you'll be emulsifying the dressing. You want to add your vegetable oil very slowly while whisking everything together. I like to measure out my oil and add it to a clean squirt bottle. But you can use a measuring cup with a spout. At first, you will want to almost add the oil drop by drop. This will take a few minutes and you will probably be left with a sore arm, but it's worth it. Just keep slowly adding oil and whisking until you're left with a smooth, thickened dressing that is thinner than mayonnaise, but still has a slightly thick consistency.
Add your Parmesan cheese and whisk to incorporate it in with everything else.
Pour dressing into a jar or container with a lid and store in the fridge for a week or so.
Chicken marinade and frying: in a medium bowl add salt and pepper, to buttermilk (add hot sauce if you want extra spice). and whisk to combine. Put two sandwich-sized pieces of chicken in a zip-top bag or bowl with a lid. Pour buttermilk marinade over the chicken and store in the fridge for at least an hour or overnight.
Make seasoned flour by combining flour with all the spices, salt, and pepper in a bowl or pan that is large enough for a piece of chicken to lay flat in the flour.
Remove one piece of chicken at a time from the marinade and dredge in the seasoned flour mixture until thoroughly coated on all sides. Place the fully coated chicken onto a piece of parchment or a sheet pan rack to rest.
Heat 2 inches of peanut oil in a skillet to 350 degrees F (175 degrees C).
Fry each piece of chicken for 6 to 7 minutes or until it reaches 165 degrees internal temperature. You'll probably want to fry for 3 minutes on the first side and then flip it to make sure you're getting the level of browning that you want. Continue cooking on the other side.
After frying, place the finished chicken on a cooling rack over paper towels to drain some of the oil away.
Garlic compound butter and toast: in a small bowl combine the butter, garlic, parsley, salt, and pepper, and using a spoon or spatula mix everything very well.
This is a garlic compound butter and it can be placed in a sealed container or wrapped in plastic wrap in the refrigerator for up to a week.
Slice your sandwich rolls and spread some garlic compound butter on the inside of each piece (about 1 to 2 tablespoons per slice).
Broil each slice for 4 to 5 minutes or until the bread is crispy and just starting to turn golden brown.
Caesar salad: while the bread is broiling, add Romaine lettuce and thinly sliced red onion to a medium bowl. Toss the lettuce mixture with Caesar dressing as you would with a salad.
Sandwich assembly: place a piece of fried chicken on top of the bottom piece of garlic roll. Top the chicken with Caesar salad.
Garnish with more parmesan cheese and/or a bit more Caesar dressing.
Close the top of the sandwich and serve.