Heat oil in a Dutch oven or large oven-safe saute pan. Add cubed chicken and season with salt and pepper. Cook, stirring occasionally, until chicken is opaque. Add spinach/kale and saute 2-3 minutes, until mostly wilted. Add garlic and saute 30-60 seconds, until fragrant.
Pour in the stock and pasta and bring to a simmer. Season with additional salt and pepper. Place a lid on the pot and cook 7-8 minutes, until pasta is al dente, stirring frequently.
Meanwhile, whisk together evaporated milk and corn starch. Once pasta is cooked, stir in milk and cornstarch mixture and allow to simmer 2-3 minutes, until thickened.
Reduce heat to low and stir in 1 cup shredded white cheese and ½ cup parmesan cheese. Season with additional salt and pepper, to taste. Top evenly with remaining white cheese, Parmesan and breadcrumbs. Bake for 15-20 minutes, until cheese is melted and bubbly.
Turn oven on a low broil and continue cooking for 2 minutes, until breadcrumbs are browned. Top with chopped parsley.