Heat 1 tablespoon oil in a 5-quart Dutch oven over medium heat. Add panko and stir for 2 to 3 minutes until golden; transfer to a dry plate and mix with 2 tablespoons Parmesan once slightly cooled.
Add remaining oil to the same pot and raise the heat to medium-high. Season chicken with salt and pepper and cook for 4 minutes, stirring occasionally.
Stir in garlic and oregano for 30 seconds. Add tomato sauce and broth, scrape the bottom and bring to a boil.
Add rotini and simmer uncovered for 12 to 15 minutes, stirring often and adding reserved water if the sauce becomes dry before pasta is tender and chicken reaches 165°F.
Remove from the heat and fold in the remaining Parmesan. Scatter mozzarella across the top, cover and rest for 3 minutes until melted.
Divide among 5 bowls and scatter the toasted breadcrumb mixture over each portion just before serving.