Preheat the oven to 400°F (204°C) and spray a 4 1/2-quart casserole dish (or a high-sided 9x13-inch dish) with cooking spray.
Cook the pasta according to package directions until al dente, drain, return to the pot, and set aside.
While the pasta cooks, heat 4 tablespoons olive oil in a large skillet over medium-high heat.
Add the diced chicken to the skillet, season generously with kosher salt, black pepper, and 1 tablespoon Italian seasoning, and cook about 5 minutes, stirring and flipping intermittently, until cooked through.
Add the minced garlic to the skillet and cook about 1 minute, stirring constantly, until fragrant.
Transfer the cooked chicken and any pan juices to the pot with the drained pasta. Add the baby spinach, artichoke hearts, and diced roasted red peppers; stir to combine and drizzle additional olive oil if the mixture seems dry. The spinach will wilt.
Taste and adjust seasoning with more salt, pepper, or Italian seasoning if desired.
Pour the pasta mixture into the prepared casserole dish, sprinkle evenly with the shredded Monterey Jack cheese, and bake about 10 minutes, until the cheese is just starting to turn light golden.
Remove the dish from the oven, sprinkle with the grated Parmesan, then return to the oven and bake another 5–8 minutes, or broil on High for up to 2 minutes until the top is golden brown (watch closely if broiling).
Optionally garnish with chopped fresh parsley and serve immediately.