Pat chicken dry with paper towels and season with salt and pepper. Heat 1 tablespoon oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add chicken in single layer and cook, without stirring, until beginning to brown, about 1 minute. Stir chicken and continue to cook until nearly cooked through, about 2 minutes; transfer to bowl and cover to keep warm.

Add 1 tablespoon oil to now-empty skillet and heat over medium heat until shimmering. Add onion and ½ teaspoon salt and cook until softened, 5 to 7 minutes. Stir in garlic, pepper flakes, and oregano and cook until fragrant, about 30 seconds. Stir in wine and simmer until nearly evaporated, 1 to 2 minutes. Stir in water, broth, and penne. Increase heat to medium-high and cook at vigorous simmer, stirring often, until penne is nearly tender, about 12 minutes.

Stir in broccoli and cook until penne and broccoli are tender and sauce has thickened, 3 to 5 minutes. Stir in chicken, along with any accumulated juice, and cook until warmed through, about 1 minute. Off heat, stir in remaining 2 tablespoons oil and Parmesan and season with salt and pepper to taste. Serve, passing extra Parmesan separately.