Different veggies cook at different times and give off different amounts of water. I prefer to make this dish one veggie at a time, and then combine them, adding the greens and seasonings at the end.
If you have two large frying pans, you can speed up the process by using them both side by side. When making veggies in batches, use about 2 teaspoons of oil per batch; mushrooms will need less. As each vegetable is done, set aside in a bowl and add additional vegetables to the bowl as they are ready.