Preheat the oven to 400 F. Coat an 8-x-10½-inch casserole dish with cooking spray.
Toss the sliced vegetables with olive oil in a large bowl. Put one-third in the bottom of the prepared dish. No need to layer them; just spread evenly. Sprinkle with one-third of the chopped chiles and season with salt and pepper. Top with half of the cheese, scattered evenly.
Repeat with another one-third of the vegetables, seasoning again with salt and pepper, roasted chile, and topping with the remaining cheese.
Finish by layering the final one-third of the vegetables and seasoning with salt and pepper and one-third of chopped chiles.
Pour milk over entire dish. Top with Parmesan cheese and sprinkle with fresh thyme. Bake, covered, for 30 minutes, then uncover and bake additional 15 minutes, until the top browns.
Remove from oven and serve warm.