Bring a large pot of salted water to a boil. Add the ziti and cook until al dente (if a range is given on the box, choose the shortest time, otherwise, cook it 1 minute less than the suggested time). Drain the pasta and return it to the pot.
Meanwhile, melt the butter with the canola oil in a large high-sided skillet over medium-high heat. Sprinkle the chicken with salt and pepper, then add to the hot skillet and cook, turning once, until golden brown and just cooked through, about 7 minutes per side. Transfer to a plate.
Reduce the heat to medium and add the cauliflower, onion and mushrooms to the skillet along with 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the onions have softened and the mushrooms are tender, about 6 minutes. Add the garlic and cook for 1 minute. Sprinkle the flour over top and cook, stirring, until all of the flour has moistened, about 1 minute. Add the wine and simmer, stirring, until reduced by about half. Add the chicken broth and 1/4 cup water, increase the heat to medium high and cook, stirring constantly, until simmering and slightly thickened. Add the heavy cream, cream cheese and 1/2 cup Parmesan; stir to combine. Pour all of the sauce and vegetables into the pot with the pasta. Add the peas, thyme and green chiles. Chop the chicken into bite-sized pieces and add to the pot. Season with salt and pepper.
Transfer the mixture to a 9-by-13-inch baking dish and sprinkle the top with the remaining 1/4 cup Parmesan. If not serving right away, let cool. Cover and refrigerate up to overnight.
When ready to bake, preheat the oven to 425 degrees F. Bake until hot and bubbling, about 45 minutes.