For the marinara: Heat the olive oil in a medium saucepan over medium heat. Add the onions and cook until softened, stirring occasionally, 8 to 10 minutes. Add the garlic and cook until softened, stirring occasionally, 3 to 5 minutes. Add the sugar, oregano and crushed tomatoes. Stir to combine and season liberally with salt and pepper. Bring to a boil over medium-high heat then reduce to a simmer. Simmer, uncovered, for 30 minutes. Discard the oregano and season additionally with salt and pepper if needed. Cover with a lid and reserve.
For the chicken: Place the chicken on a cutting board. Place one hand firmly on top of one of the chicken breasts and carefully cut in half horizontally with a sharp chef's knife. Repeat with the other chicken breast. Sprinkle the chicken liberally with salt and pepper on both sides. Place the flour, eggs and breadcrumbs into three separate shallow baking dishes. Dredge each of the chicken pieces first in the flour, coat with the beaten eggs, and finish in the breadcrumbs. Place the dredged chicken pieces onto a plate as you go. Heat 1/4 cup of the olive oil in a large saute pan between medium and medium-high heat. Add 2 of the dredged chicken pieces and cook until browned on the first side, 2 to 3 minutes. Flip and cook until the second side is browned, 2 to 3 minutes. The chicken does not need to be fully cooked at this point because it will finish cooking in the oven. Transfer the browned chicken to a paper-towel-lined plate to absorb any excess oil. Season lightly with salt and pepper. Discard the olive oil from the pan and wipe out the pan with a dish towel. Pour the remaining 1/4 cup olive oil in to the pan and repeat cooking process with the remaining chicken. Rest the cooked chicken for 5 minutes. Cut the chicken diagonally into 1-inch slices, 4 to 5 pieces per half of chicken. Set aside.
For the rolls: Preheat the oven to 400 degrees F with a rack on the bottom shelf. Place a large baking sheet on the rack. Divide the dough into 4 equal pieces. On a lightly floured flat surface, stretch or roll each piece of dough into a 9- by 7-inch rectangle, about 1/8-inch thick.
With the longer side of the dough facing you, pile 1/4 cup mozzarella across the middle of the dough, leaving a 1/2-inch border on the left and right sides. Sprinkle on 1 tablespoon Parmesan, then place one-quarter of the chicken strips lengthwise. It is okay if the chicken is peeking out of the dough slightly. Spread 1/4 cup marinara sauce on top. Finish with 1/4 cup mozzarella and 1 tablespoon Parmesan. Fold the piece of dough that is closest to you over the filling. Brush 1/2-inch of the dough with the egg wash. Fold the remaining dough over the filling, making sure that the two ends overlap where the egg wash is brushed. Pinch the ends together to seal securely. Carefully flip over the roll so that the seam is on the bottom. Brush the top and sides with the egg wash. Make the remaining three rolls using the rest of the ingredients. Remove the hot baking sheet from the oven and lightly brush all over with canola oil. Carefully place the chicken rolls on the baking sheet, making sure that there is space between each roll. Bake until golden brown and cooked through, 35 to 40 minutes, rotating the baking sheet halfway through. Remove from the oven and cool on the baking sheet for 10 minutes.
To serve, reheat the marinara over low heat, adding 1/2 cup water to thin the sauce slightly. Serve with the hot chicken rolls.