Start by melting butter in a large skillet over medium heat. Add your thinly sliced onions and stir until coated. Let them cook slowly for 30 minutes, stirring occasionally, and deglaze with chicken stock to lift all the caramelized bits from the pan.
Dust each cutlet with flour and season with salt and pepper. Dip them in beaten egg, then coat with the breadcrumb and parmesan mixture. Chill in the fridge for 10 minutes to let the coating set for perfect frying.
Heat canola oil in a skillet over medium-high heat. Fry the chicken cutlets for 3-4 minutes on each side until golden brown and cooked through. Drain on paper towels and season with kosher salt while still hot.
Preheat your oven to 400˚F. Slice your rolls and layer with marinara, a chicken cutlet, more sauce, and a few caramelized onions. Top with two slices of burrata or mozzarella and bake for 5-10 minutes until cheese melts and bubbles.
Remove from the oven and drizzle chimichurri over the sandwiches. Serve immediately while the cheese is warm and gooey.