Make the giardiniera- add the cauliflower, pepper, shallot, jalapeno, and carrot to a bowl and add in the vinegar and salt. Add some water until it covers the veggies. Refrigerate overnight. Strain the mixture and discard the liquid. Put the veggies in a new bowl with the olives and parsley and stir in the garlic. Drizzle with olive oil. Cover and store in fridge until ready to use.
Make the chicken - I had very large chicken breasts so I used one breast for 4 cutlets. I first cut the chicken in half, then sliced each piece in half sideways to make thinner pieces. Put some parchment on the work surface and add a piece of chicken. Top with another piece of parchment and gently pound with a kitchen mallet or rolling pin. You want these to be very thin. Repeat with remaining chicken pieces.
Add the flour to one large bowl, the eggs to another, and the breadcrumbs and parm in a third. Season the chicken with salt and pepper. Dredge the chicken in the flour, then the eggs, and then coat with the breadcrumb mixture. Press the breadcrumbs into the chicken to make sure it is fully coated. Place on a plate and repeat with remaining chicken.
Add some oil to a pan so it is a thick coating, about 1/4 inch deep. Bring to medium high heat. Shallow fry the chicken on each side to golden brown. Set aside on a rack.
Cut the Italian bread into bun sized pieces. Melt the butter and stir in the oregano, grated garlic, and red pepper flakes to your personal taste. Brush the mixture on the pieces of bread.
Warm up the tomato sauce in a pan or in the microwave.
Preheat broiler to high. Broil the buns to golden brown. Top the chicken pieces with cheese, some with mozzarella and some with provolone. Broil to melt the cheese.
Build the sandwich - bottom bun first, then a piece of chicken with mozzarella, then sauce, then chicken with provolone, then more sauce, then a heaping spoon of giardiniera. Add the top bun and serve.