Make the tomato sauce. Add the olive oil and half of the butter to a saucepan over medium heat. Once the butter is melted, add the grated onion and garlic and cook for 2-3 minutes, until the onion is soft. Stir in the oregano and tomato paste and cook for another minute. Stir in the crushed tomatoes, sugar, and season the sauce with salt and pepper. Reduce the heat to low and simmer for 20 minutes. Stir in the rest of the butter and season to taste, as needed. Let the sauce cool. You can make it ahead of time and store in the fridge for up to 1 week in an airtight container.
Prep the chicken. Use a sharp knife to butterfly the chicken breasts so that there are 4 chicken breast pieces of even thickness. If your chicken breasts are small, you can use a flat mallet to pound them into an even 1/2” thickness, in which case you'll want to use 4 small breasts. Pat the chicken very dry all over with paper towels and season lightly with salt and pepper on both sides.
Batter & bread the chicken. In a medium mixing bowl, whisk together the eggs, flour, Calabrian chile, and water with a bit of salt and pepper until smooth. Pour the panko, parmesan, and oregano into a shallow dish with some salt and pepper and mix well. Dip the chicken in the flour and egg batter until they’re well coated, shaking off any excess. Coat well in the panko breadcrumbs and set on a plate or tray. Repeat with the remaining chicken.
In a high-sided skillet or braiser, pour in enough canola oil to come halfway up the chicken–about 1/2 inch of oil in the skillet. Heat over medium-high heat, until the temperature reaches 350 F. Meanwhile, preheat an oven to 400 F.
Fry the chicken. Once the oil is hot, gently lower the chicken into the oil taking care not to overcrowd the pan. Cook 3-4 minutes per side, or until the panko coating is golden brown & the chicken is mostly cooked through. Once both sides are golden brown, transfer the fried chicken to a paper towel-lined plate to drain excess oil. Season with a good pinch of flaky salt while the chicken is still warm. Repeat with the remaining chicken.
Finish the chicken in the oven. Transfer the chicken to a parchment lined baking sheet. Top each piece of chicken with about 1/3 cup of sauce and top with 1/2 cup of the shredded cheese. Transfer to the oven for 10 minutes, then broil the chicken on high for 2-3 minutes, until the cheese is bubbling and browned. Top the chicken with chopped parsley and serve right away.