Preheat the oven to 375 degrees F.
Bring a large pot of salted water to a boil, then add the pasta. Boil, and remove from the water about 2 minutes under what the package directions state as al dente. You want the pasta to be a bit undercooked because it will finish cooking in the oven. Once the pasta is about 2 minutes from al dente, from the pot using a skimmer, and transfer to a 9x13-inch baking dish. Do not rinse the pasta, and set aside about 1 cup of pasta water before returning the water to a boil (don't drain the water!)
Once the water returns to boiling, add the broccoli and cook for 2-3 minutes until starts to soften but still bright green. Using the skimmer, remove the broccoli and place it into the baking dish with the pasta. Set aside.
Meanwhile, heat a large skillet over medium-high heat. Season the chicken with 1.5 teaspoons of kosher salt, 1 teaspoon of garlic powder, 1 teaspoon of onion powder, 1 teaspoon of oregano, about a ½ teaspoon of black pepper (or to taste), and about ¼ teaspoon of red pepper flakes (optional).
Add 2 tablespoons of olive oil, to the hot skillet, then add the chicken and sear until brown on both sides, about 3-5 minutes per side. It's okay if they aren't cooked all the way through, they will continue cooking in the oven. Remove from the skillet and set aside to rest for a few minutes before cutting them into bite-sized pieces. Set aside.
In the same skillet, reduce the heat to medium, and add 2 tablespoons of butter, followed by the onion with a pinch of salt. Cook until softened and translucent, about 4-5 minutes, then stir in the garlic, the remaining 2 teaspoons of oregano, pepper, and red pepper flakes to taste. Cook, stirring constantly to prevent burning, until the garlic is fragrant, about 30 seconds to 1 minute, then immediately deglaze the pan with chicken broth.
Reduce the heat to low, then add the cream cheese in chunks, and stream in 4 ounces of heavy cream. Simmer for 3-4 minutes, stirring often to ensure nothing is sticking on the bottom, until the mixture is thick yet pourable. Slowly stream in some of the pasta water if needed to thin.
Reduce the heat to low, and mix in about half of the cheeses in small handfuls stirring to melt before adding more cheese in. This (2.5 ounces) of the parmesan, ½ of the mozzarella (about 4 ounces), and ½ (about 4 ounces) of the Colby jack. Make sure to reserve the rest for topping.
Stream in reserved pasta water if needed to thin the cheese sauce, then add in the chicken pieces. Adjust the seasonings to your taste.
Pour the cheesy chicken mixture over the pasta and broccoli in the baking dish and stir to combine. Sprinkle the remaining cheese over top.
Bake uncovered for about 25-30 minutes, until the cheese is lightly golden brown and the sauce is bubbling. Enjoy!
Store leftovers in an airtight container or cover it in foil once it has cooled down. Place it in the fridge for up to 4 days and warm portions in the oven or microwave when needed. If freezing, take note that the sauce will seize up when frozen and might become grainy when reheated. You can place the pasta in a freezer-friendly container for up to 2 months.