Step 1Cut coarse stems off spinach. Place leaves in colander and rinse with cold water, lifting and tossing spinach to wash away grit. Shake spinach to get rid of excess water and put it in a large bowl. Set aside.
Step 2Cut off stem top and root bottom of each zucchini. Slice in 1/4-inch slices and add to spinach.
Step 3Cut stem top and root bottom off onions. Cut onions in half lengthwise from stem top down. Put each half flat side down on a cutting board and cut lengthwise in strips 1/4-inch wide, then holding strips together, cut onion in 1/4-inch strips crosswise. Add onions to bowl.
Step 4Stack basil leaves in a pile, and with sharp paring knife, chop into small pieces. Add to bowl. Using your hands, reach to bottom of bowl and gently toss and turn vegetables until they are well mixed. Season with salt and pepper to taste and turn to mix well.
Step 5Place vegetable mixture in 4-quart casserole and drizzle oil evenly over top. Cover and bake at 350 degrees, checking after 30 minutes to see if all greens seem tender and nicely blended. If they are, sprinkle cheese over top, put lid back on and bake another 5 minutes. If they aren't done, bake another 5 to 10 minutes before adding cheese. Serve hot.